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Flow characteristics of fluid-solid mixtures Johnson, Norman Allan 1967

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PLOW CHARACTERISTICS OP FLUID-SOLID MIXTURES by NORMAN ALLAN JOHNSON B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1967 i In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes other than p u b l i c a t i o n may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying any part of t h i s thesis f o r p u b l i c a t i o n s h a l l not be allowed without my written permission. Department of Mechanical Engineering, The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, B r i t i s h Columbia. A p r i l , 1967-ABSTRACT This thesis deals with pneumatic conveying i n a closed duct. The closed duct consists of approximately 260 feet of 2-inch standard pipe made up of h o r i z o n t a l , v e r t i c a l , and elbow sections. An expression i s developed which provides the pressure d i f f e r e n t i a l between the two ends of the closed duct i n terms of the variables which appear to a f f e c t two-phase flow. The development and assembly of the apparatus used i n studying and evaluating the theory i s des-cribed i n considerable d e t a i l , with a view to future work i n t h i s f i e l d . The data acquired from the experimental apparatus i s analysed i n an attempt to co r r e l a t e the various flow resistance f a c t o r s , due to the p a r t i c u l a t e phase, with the c h a r a c t e r i s t i c s of the p a r t i c u l a t e m a t e r i a l . The r e s u l t s and conclusions derived from the experimenta-t i o n and analysis described herein, while not d e f i n i t i v e , shed some new l i g h t on the complexities of two-phase flow and provide a r a t i o n a l empirical approach to the study of two-phase flow systems. i i i TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION . . . . . . . . . . . 1 2 LITERATURE REVIEW 5 Generalized Approach . . . . . . . . . . . . 5 Plow Characteristics . . . . . . . . . . . . 10 Velocity Characteristics . .> 12 Pressure Gradients , . . . 15 Thermodynamic Effects . . . . . . . . . . . 16 Particle Characteristics 17 Summary . . . . . 17 3 ANALYSIS . . . .;';=...v;,.l- 19 Description of Plow . . . . . 19 Basic Equations . . . . . . . 21 Simplifications for Pneumatic Conveying . . 27 The Elimination of V n in Equation 3(13) 27 Determination of Pressure as a Function 0-f X o o © o o o o o o 6 « e e • o • • 2 3 Use of Basic Gas Laws . . 29 Calculation of F for Straight Ducts . . 31 Calculation of F§ for Ducts with Constant Radius of Curvature 32 S l i p Ratios for Any Duct 36 Selection of a Function to Describe C D and f^ , i n Terms of Reynolds Number . . 40 Integration of Equation 3(13) for Pneumatic COHV@yXHg o o o o o o o b o o o o o o o e S\mmi£Li*y o o o « o o o o o o o o » o o o » o ^9 4 COMPARISON OF THE THEORY WITH THE WORK OF OTHERS 51 Consideration of Slip Ratios . . 51 Determination of Velocity for Minimum Pressure Drop i n a Duct . 56 Summary . . . . . . . . . . . . . 5^ 5 EXPERIMENTAL APPARATUS 59 The Duct .« . . . • . . . . . 60 The Blower . • . . . • . 63 The Injector . . . . . ;.' . ° . . . .v . . . 73 Driving the Rollers . . . 75 A l l * XjOSS€S o o o o o o e o o o o o o o o o 7^ Injection Characteristics 81 Receiver, Supporting Tower & Collector Chute 89 Instrumentation . . . . . . . . . . . . « • 92 Summary o o « . o . • . o 0 . . . 0 o o . . . . . « 97 i v CHAPTER PAGE 6 EXPERIMENTAL PROCEDURE 98 Proposed Method of Data Analysis 98 Selection of Particulate Materials 100 Selection of Variables for Measurement . . . 101 Sequence of Operations 108 Summary . . . . . . . . . . 110 7 ANALYSIS OF EXPERIMENTAL RESULTS . . . . . . . 112 Pressure Gradients 113 Computations 119 Air Properties and Plow Characteristics 120 Solution of the Three Sets of Ten Simultaneous Equations for f 0 , f z and f, 2. A . . 124 Variation of f 0 with Air-to-Material Ratios . 126 Slip Ratios and the Evaluation of 0 D . . 130 Summary . . . . . . . . . . . . . 136 8 CLOSURE . . . . 137 Recapitulation 137 Conclusions 139 Recommendations for Future Research . . . . 143 Summary . . . . . . . . . . . . 147 BIBLIOGRAPHY . . . . . . . . . . . . . 148 APPENDIX I Tabulated Experimental and Computed Data . . . 151 APPENDIX II Computations and Computer Program 181 Fortran Program 185 V LIST OF FIGURES FIGURE PAGE 1 Velocity Versus Pressure Drop per Unit Length. 6 2 Velocity Versus Pressure Drop for Various Mustard Seed Loadings . . . . . . 11 3 L i f t Forces on a Single Particle 21 4 Two-phase Flow in a Straight Duct 23 5 Resistance Offered to a Single Particle . . . . 32 6 Curved Duct i n Cartesian Co-ordinates . . . . . 33 7 Accuracy of the Force-Fit Function for C D . . . 43 8 Variation of Material Velocity with Air Velocity 55 9 System Schematic and Instrumentation 61 10 Blower, Silencer and Motor . . . . . . . . . . 65 11 Accuracy of Function used to Compute V & over Pertinent Range 70 12 Blower Discharge Rate Related to Blower Speed and Discharge Pressure . . . . . . . . . . . . 71 13 Approximate Velocity Characteristics . . . . . . 72 14 Injector Installed . . . . . . . . . . . . . . . 74 15 Feeder Efficiency versus a Measurable Pressure Parameter . . . . . . . 80 16 Cross-Section Through Injector . . . . . . . . . 83 17 Log-Log Plot, of Injection Parameters . . . . . . 85 18 Cyclone Receiver at the End of the Duct . . . . 88 19 Supporting Tower and Receiver . . . . . . . . . 90 20 Manometer Arrangement . . . . . . . . 93 21 Veigh-Barrel and Strain-Gage Ring . . . . . . . 95 V I FIGURE PAGE 22 Approximate Linearity of Pressure Gradients 114-23 Effects of Non-Positive Displacement Blower 115 24- Pressure Gradient Variations at Injector, Elbows and Vertical Sections of Duct . . . . 118 25 Correlation of Polytropic Gas Constant with Air-to-Material Ratio . 123 26 Variation of C^/f^ with W /VT, for Various D' 2 ac' m Combinations of f^ and f ^ . . . . . . . . . 127 27 Variations of fVC,, with W /W Q„ for one ' d' D m ac Combination of f^ and f ^ . 129 28 Experimental values of v„„„ and v„„„ 132 r aav mav 29 Average S l i p Ratios versus Air-to-Material Ratio 133 30 Average Slip ratios versus Material-to-Air SlSL"fclO S » « . o o « o o o « o o o e o o o o « ^ 31 Hypothetical Variations of f 2 and C D with Material-to-Air Ratio 135 v i i LIST OF TABLES TABLES IN TEXT PAGE I The Slip Ratios of Hitchcock and Jones . . 13 II S l i p Ratios of Richardson and McLeman . . 15 F r i c t i o n Factors Calculated from Literature Data ....... . . . 56 IV Particulate Materials Tested 101 V Variables Measured During Test-Runs . . . 107 TABLES IN APPENDIX I A Apparatus Component Specifications . „ . 151 B Legend of Duct Distances . . 152 P.. Blower Characteristic Data 153 D Injector Air-Loss Data . 154 E Injector Feed-Rate Data . <> 155 p i . - F3 Data from Test-Runs 156 - 158 G Pressure Gradient Readings from Test-Run #1 159 H1 - H3 Air Properties and Flow Characteristics 160 - 162 J1 - J3 Experimental Slip Ratio Data , . . o . « . 163 - 165 KT - K10 Calculated Values of f 2 and Slip Ratios. . 166 - 175 M1 - M3 Ratios of Polytropic Constants and Flow- 176 - 178 N P a r t i c l e Drag C o e f f i c i e n t and Reynolds Number Ratios . . . . . . . . . . . 179 v i i i LIST OF VARIABLE SYMBOLS SYMBOL DESCRIPTION DIMENSIONS A A T d D D. F F' g h L L A L r m Area L Cross-sectional area of the duct L c Cross-sectional area of a mean p a r t i c l e Lc Drag C o e f f i c i e n t Diameter of a mean p a r t i c l e L Diameter of the Duct L Diameter of the flow-measurement o r i f i c e L Darcy-Weisbach f r i c t i o n f a c t o r i n single phase f l u i d flow Relative flow f r i c t i o n f a c t o r due to flow of f l u i d past p a r t i c l e s S l i d i n g f r i c t i o n f a c t o r f o r p a r t i c l e s on duct walls Flow f r i c t i o n f a c t o r due to the random motion of p a r t i c l e s Total force r e s i s t i n g flow of p a r t i c u l a t e phase MLT Volumetric feed rate per re v o l u t i o n of , i n j e c t o r r o l l s I> A c c e l e r a t i o n due to g r a v i t y LT Material head over i n j e c t o r r o l l e r s L Equivalent length of duct L Actual h o r i z o n t a l length of duct L Actual v e r t i c a l length of duct L Actual t o t a l length of duct,including elbows L Length of i n j e c t o r r o l l e r s L Counter i n summations -2 -2 i x Polytropic gas constant - -Number of items, e*go particles -1 Blower speed, rpm T =•1 Injector speed, rpm T Number of straight sections i n a generalized duct Pressure, usually subscripted ML" T Internal pressure of injector r o l l e r s ML T Pressure drop over length of duct during either single- or two-phase flow ML T Number of curved sections i n a generalized duct Volumetric flow rate L-'T Radius of curved ducts or elbows L Radius of injector r o l l e r s L Gas constant L D e g Reynolds Number with respect to the duct Reynolds Number with respect to the particles Hydraulic radius L S l i p r a t i o , v m/v a, usually subscripted Specific gravity of particles =» 2 Particle surface area 2j Spread of injector r o l l s L Time required to empty the hopper over the injector H Temperature Deg Relative velocity of air and particles LT° Velocities of phases with respect to the duct LT~ X V, Volume of material in hopper over injector , at steady-state Ir -2 w Weight of a mean particle MLT W Weight? flow-rate of phases MLT"5 -2 W Material weighed i n the weigh-harrel MLT x The distance to a point on the duct from the center of the injector L oC Angle of inclination of duct ^ Injector a ir loss factor or efficiency © Angle of tend of a curved duct jCC Viscosity MT"1L~1 ? Density ML~2T~2 SUBSCRIPTS SYMBOL DESCRIPTION a Fluid phase i n two-phase flow at Atmospheric conditions av Average values b Bulk characteristics of particulate phase c Curved duct; calculated value; or corrected value D Relating to the duct E Relating to the elbow or curved section(s) of a generalized duct f Free-falling characteristics or f i n a l velocity H Relating to the horizontal portion(s) of a generalized duct m Flowing particulate phase or a counter i n summations mx Maximum value of variable o Free-running conditions p Relating to individual particles s Straight duct x Linear distance to a point from the center of the injector V Relating to the ve r t i c a l sections of a generalized duct 1 * 2 , . . , 21 Points defined by pressure taps i n duct x i i ACKNOWLEDGEMENTS The author wishes to express h i s appreciation f o r the assistance he received from the Mechanical Engineering Laboratory Technicians, the Department Secretaries, the Computing Center S t a f f , and Dr. J.L. Wighton. In a d d i t i o n , c r e d i t i s due the National Research Council and J.A. Beckow Limited f o r the f i n a n c i a l assistance which made t h i s research possible. CHAPTER 1 INTRODUCTION This thesis i s concerned with that branch of two-phase flow commonly known as pneumatic conveying. The design of a pneumatic conveyor entails the determination of the i n i t i a l velocity, pressure, and quantity of air and the size of duct required to transport a given amount of particulate material through a given duct configuration. Normally, the i n i t i a l a i r velocity i s that required to prevent saltation i n the duct and i s determined experimentally for any given material. The i n i t i a l velocity and the duct diameter determine the quantity of a i r required. Consequently, what i s usually sought i n con-veyor design i s an optimum combination, from a power consump-tion standpoint, of duct size and pressure drop across the duct configuration. The two-phase flow process involved i n pneumatic convey-ing might be described as follows: solid particulate material i s propelled along a closed duct by air at a velocity s i g n i f i -cantly less than that of the a i r ; the pressure d i f f e r e n t i a l between the two ends of the duct requires that the air expand and accelerate along the duct, similarly accelerating the par-ticulate material; an energy exchange between the duct and i t s environment may occur due to heat transfer; a continual energy 2 exchange occurs "between the two phases, to some extent due to heat t r a n s f e r , hut p r i m a r i l y through dynamic processes i n v o l v i n g c o l l i s i o n s between p a r t i c l e s , between p a r t i c l e s and the duct w a l l , and between p a r t i c l e s and a i r molecules. Almost a l l the energy required to maintain a two-phase process of t h i s type i s d i s s i p a t e d i n the o r i g i n a l a c c e l e r a t i o n of the f l u i d and p a r t i c u l a t e phases, through the f r i c t i o n a l e f f e c t s associated with the flow of a single homogeneous f l u i d , and through f r i c t i o n a l e f f e c t s associated with the flow of the s o l i d p a r t i c u l a t e phase. Among the many va r i a b l e s which enter into the optimiza-t i o n of pressure drop and duct s i z e are p a r t i c u l a t e material flow r a t e , bulk density, and cohesiveness; p a r t i c l e s o l i d density, size d i s t r i b u t i o n , mean s i z e , shape and roughness; a i r losses at the i n j e c t i o n point or at a u x i l i a r y equipment along the duct; a i r density and v i s c o s i t y ; duct roughness, f r i c t i o n f a c t o r , and geometric configuration; and atmospheric conditions. Many researchers and entrepreneurs have attempted to cor r e l a t e the above v a r i a b l e s . A b r i e f look at these attempts, as published i n the l i t e r a t u r e of the f i e l d , i s presented i n the next chapter. However, by way of intro d u c t i o n , i t might be worth s t a t i n g that a l l of the l i t e r a t u r e reviewed i s con-cerned with s t r a i g h t ducts, p r i m a r i l y h o r i z o n t a l i n o r i e n t a t i o n , and ignores the e f f e c t s of co m p r e s s i b i l i t y of a i r . Most of the 3 l i t e r a t u r e deals with sp h e r i c a l or nearly s p h e r i c a l p a r t i c l e s . Experimental data are most frequently co r r e l a t e d on an empiri-c a l "basis. C o r r e l a t i o n "between experimental r e s u l t s and those t h e o r e t i c a l considerations which do e x i s t leaves much to he desired. The need f o r such a c o r r e l a t i o n l e d to the research presented i n t h i s t h e s i s . Hence, the problem and the method used i n i t s s o l u t i o n may now be stated. The purpose of t h i s research was to provide a r a t i o n a l and experimentally v e r i f i a b l e r e l a t i o n s h i p between the many v a r i -ables i n f l u e n c i n g a two-phase flow process and the pressure drop along the duct i n which the process occurs. In scope, the research was r e s t r i c t e d to pneumatic conveying. The method of attack consisted of f i v e basic steps. F i r s t , a large segment of the pertinent l i t e r a t u r e of the f i e l d was studied with a view to determining, what others might have succeeded i n doing toward r a t i o n a l i z i n g the many var i a b l e s which govern pneumatic conveying. Second, with reference to the l i t e r a t u r e , a theory was postulated whereby the pressure d i f f e r e n t i a l required i n any a p p l i c a t i o n i s ex-pressed i n terms of as many of the va r i a b l e s involved as po s s i b l e . T h i r d , an apparatus was devised, c o n s i s t i n g of approximately 260 feet of two-inch diameter pipe arranged such that there were four h o r i z o n t a l runs, two v e r t i c a l runs, and f i v e three-foot radius elbows; an approximately constant d i s -placement blower; a novel i n j e c t i o n apparatus which provided 4 a steady rate of feed as opposed to the unsteady rate of feed inherent i n a pocket-rotor feeder; a cyclone receiver located over the injection apparatus such that continual c i r -culation of the particulate material was maintained under steady-state conditions; and instrumentation consisting of mercury and water manometers, thermocouples, and a strain-gage weighing apparatus. Fourth, tests were conducted, using plastic pellets and barley grains, to study and verify the theory and to glean as much information as possible from the experimental data about those factors not adequately covered by the theory but seemingly important to the design of pneu-matic conveyors. F i n a l l y , the results were carefully anal-ysed, both manually and with the aid of an IBM 7040 computer. CHAPTER 2 LITERATURE REVIEW I t was noted i n the Introduction that many attempts have been made to co r r e l a t e the v a r i a b l e s inherent i n two-phase flow. In the case of pneumatic conveying, t h i s i s p a r t i c u l a r l y true. Let us consider some of the better examples of t h i s l i t e r a t u r e . GENERALIZED APPROACH From a t h e o r e t i c a l standpoint, an attempt to corr e l a t e the v a r i a b l e s involved i n f l u i d i z a t i o n , p a r t i c l e systems, hydraulic conveying, and pneumatic conveying has been made re c e n t l y by F.A. Zenz and D.F. Othmer i n t h e i r text e n t i t l e d 1 F l u i d i z a t i o n and F l u i d - p a r t i c l e Systems. Chapter ten of t h i s text deals with pneumatic and hydraulic conveying. Most of the papers on these subjects published by Zenz up to 1960 are contained i n t h i s chapter. Among many other things, he attempts to co r r e l a t e s u p e r f i c i a l f l u i d v e l o c i t y with pressure d i f f e r e n t i a l f o r a f i x e d p a r t i c l e flow r a t e . Another aspect of Zenz 1 work r e l a t e s to optimum v e l o c i -t i e s f o r minimum pressure drops i n both h o r i z o n t a l and v e r t i c a l flow. The general shape of the c h a r a c t e r i s t i c curve f o r v e r t i c a l flow i s shown i n Figure 1. 6 Published i n the same month as Zenz published his findings regarding velocities for minimum pressure drop, p June 1957» was a paper by BLEo Rose et a l which contained much of the same information i n a s l i g h t l y different form; reference to this paper i s made a l i t t l e later» LOG SUPERFICIAL F L U I D V E L O C I T Y , MS F i g „ 1 Velocity Versus Pressure Drop Per Unit Length. F r i c t i o n i n the Flow of Suspensions^, by E.G. Vogt and H„R. White i s one of the most frequently cited papers i n the literature of the f i e l d . This paper presents the c o r r e l a t i o n where A and K are given as empirical functions of the dimen-7 s i o n l e s s group Although, t h i s c o r r e l a t i o n can presumably be used f o r h o r i -zontal and v e r t i c a l ducts, no consideration of the e f f e c t s of elbows and p a r t i c l e c h a r a c t e r i s t i c s i s given. h. Designing a Pneumatic Conveyor , by S.J. M i c h e l l , leans h e a v i l y on the work of Vogt and White, although i t o f f e r s a s l i g h t l y d i f f e r e n t method f o r c a l c u l a t i n g A and k. In addition, M i c h e l l ' s p u b l i c a t i o n o f f e r s a method f o r determining the "safe" a i r v e l o c i t y f o r conveying p a r t i c l e s over 100 microns i n d i a -meter. E s s e n t i a l l y , t h i s method e n t a i l s the use of the formula V e l o c i t i e s below t h i s value apparently threaten to r e s u l t i n s a l t a t i o n and plugged pipes or ducts. On The Pressure Drop and Clogging Limits i n the H o r i -19 zontal Pneumatic Conveyance Pipe 7 , by Shinzo Kikkawa et a l , i s one of the most i n t e r e s t i n g of the recent publications i n t h i s f i e l d . The authors started with the f a i r l y common assumption that the pressure drop f o r two-phase flow i s com-posed of two separate pressure drops. Hence, AP = AP + £P a m 8 They found that the pressure drop due to the p a r t i c u l a t e phase could be expressed as follows; m 2 D W a where \ a is c a l l e d the " c o e f f i c i e n t of pressure drop due to s o l i d p a r t i c l e s " o Prom the f i r s t p r i n c i p l e s of dynamics, an expression i s derived whereby A i s given i n terms of p a r t i c l e s p r o p e r t i e s , a f r i c t i o n f a c t o r which represents the only r e s i s t -ance the p a r t i c u l a t e phase encounters i n the duct, and the somewhat nebulous term, ~\g, which i s "the duration (time) of s o l i d p a r t i c l e staying i n the conveyance pipe". No attempt i s made to c o r r e l a t e ^ s with p a r t i c l e p r o p e r t i e s . v a Choosing the Proude number, ^g", as t h e i r independent v a r i a b l e , the authors have p l o t t e d many curves showing \ v e r s u s ^ g g - f o r various values of p a r t i c l e f r e e - f a l l i n g or terminal v e l o c i t y , f r i c t i o n f a c t o r , and m a t e r i a l - t o - a i r r a t i o . They noted that a f r i c t i o n f a c t o r of 1.0 i s equivalent to the resistance a p a r t i c l e would encounter i n a v e r t i c a l duct. However, no consideration i s given to curved ducts or to ducts of generalized configuration. Unfortunately, t h e i r experimental r e s u l t s do not correspond very well with t h e i r t h e o r e t i c a l curves. On the other hand, the comparison the authors draw between t h e i r own t h e o r e t i c a l analysis and the empirical work of others, such asVogt and White, suggests very strongly that t h e i r own approach i s one of the best taken to date and i s a 9 major advancement i n the quest f o r a r a t i o n a l understanding of two-phase flow. Several papers and books view pneumatic conveying from a p r a c t i c a l standpoint. Wilbur G. Hudson, i n Conveyors and  Related Equipments presents an empirical approach to the design of pneumatic conveyors, which i s used by a large and reputable manufacturer of pneumatic conveying equipment. This approach centers mainly around the Unwin empirical formula f o r pressure loss i n pipe f o r a i r alone, AP_ = 1.12v?L(D + 5.60) x 1CT 6psi. D Hudson determines material pressure losses on the basis of energy r e l a t i o n s , empirical constants f o r d i f f e r e n t p a r t i c u l a t e materials, and an empirical r e l a t i o n f o r the s o - c a l l e d "balancing v e l o c i t y " . Another general and p r a c t i c a l t r e a t i s e on the f i e l d i s Pneumatic Handling of Powdered Ma t e r i a l , published by the E.E.TJ.A. i n B r i t a i n . This book contains much of the same i n -formation and i s j u s t as oriented toward the empirical design of pneumatic conveyors as Hudson's book. However, i t i s a l i t t l e more current, r e f i n e d , and comprehensive i n i t s t r e a t -ment of the p r a c t i c a l engineering aspects of system design. 10 FLOW CHARACTERISTICS A v i v i d d e s c r i p t i o n of three "basic types of two-phase flow i s given i n A.H. Korn's paper "How S o l i d s Flow i n Pneumatic  Handling Systems^. He suggests that, " S o l i d p a r t i c l e s may advance by p r o g r e s s i v e l y increasing leaps i n large pipes; i n suspended condition i f l i g h t and acted on by steep v e l o c i t y gradient; or suspended as true f l u i d i f f i n e enough to be a f f e c t e d by Brownian Movement". He implies that a steep v e l o c i t y gradient between the outside and center of the pipe w i l l create l i f t forces which tend to draw a l l p a r t i c l e s toward the center of the pipe. Leonard Farbar, i n h i s paper, Flow C h a r a c t e r i s t i c s of Q Solid-Gas Mixtures , o f f e r s several p l o t s of a i r and material flow rates versus pressure drop. He concludes by saying that p a r t i c l e s i z e d i s t r i b u t i o n s i g n i f i c a n t l y influences flow c h a r a c t e r i s t i c s , although he does not s p e c i f y how, and h i s curves are somewhat d i f f i c u l t to i n t e r p r e t . 2 I t was noted above that.H.E. Rose et a l publisiied a general t r e a t i s e on pneumatic conveying which was r e l a t e d to Zenz* optimum v e l o c i t y , or s a l t a t i o n v e l o c i t y work. Rose s p e c i f i e s , i n Flow of Suspensions of Non-Cohesive Spherical 2 P a r t i c l e s i n Pipes , that the optimum v e l o c i t y , as regards pressure drop, v a r i e s according to the geometric configuration of the conveying duct. This optimum v e l o c i t y increases as the 1 1 angle of i n c l i n a t i o n i s increased upward from the h o r i z o n t a l . A t y p i c a l set of curves, presented here f o r reference l a t e r , i s shown i n Figure 2. V* , -f+/sec Vk , fVsec One-Inch Horizontal Duct One-Inch V e r t i c a l Duct Fig . 2 V e l o c i t y versus Pressure Drop f o r Various Mustard Seed Loadings. Among Rose's conclusions are several of s i g n i f i c a n c e to t h i s t h e s i s . " 1 . The t o t a l pressure d i f f e r e n t i a l f o r conveyance of the suspension can be expressed as the sum of three component pressure drops, namely, a pressure drop of the conveying f l u i d , a f r i c t i o n a l pressure drop due to the presence of the s o l i d s and a s t a t i c pressure drop also due to the presence of s o l i d s . 12 The pressure drop of the f l u i d may be cal c u l a t e d from any of the pipe flow formulae applicable to a homogeneous f l u i d . The s t a t i c component may be ca l c u l a t e d i n the normal manner f o r f l u i d s provided that the density of the suspension i s used i n place of the density of the f l u i d . The density of the suspension can be expressed most conveniently i n terms of the weight rates of flow of the s o l i d and f l u i d parts of the suspension. The f r i c t i o n a l pressure drop, due to the presence of the s o l i d s , i s independent of the slope of the pipe, and of the siz e of the s o l i d p a r t i c l e s , f o r values of d/D ranging from 0.007 to 0.24-5. The f r i c t i o n a l pressure drop due to the presence of the s o l i d s i s a func t i o n based on the Reynolds number f o r the flow of the homogeneous f l u i d i n the pipe and may be r e l a t e d to that Reynolds number by a f r i c t i o n f a c t o r . . . . " VELOCITY CHARACTERISTICS Obviously, the p a r t i c u l a t e material i n a duct must t r a v e l at a lower v e l o c i t y than the transporting f l u i d . The r a t i o of the p a r t i c l e v e l o c i t y to the f l u i d v e l o c i t y i s often r e f e r r e d to i n the l i t e r a t u r e as the v e l o c i t y " s l i p " r a t i o or f a c t o r . This term i s used i n the same sense i n t h i s t h e s i s . The work of Zenz i n the area of s a l t a t i o n v e l o c i t i e s has already been mentioned. S p e c i f i c a l l y , h i s paper, Minimum  V e l o c i t y f o r Catalyst Elow^, was the source of Figure 1. Nowhere i n Zenz 1 w r i t i n g s , at l e a s t to the knowledge of t h i s w r i t e r , i s any mention made of the seemingly constant rate of change of the s l i p r a t i o with increasing a i r v e l o c i t y . Several of the c o r r e l a t i o n s i n the above paper are a l i t t l e incomprehensible. However, the work r e l a t e d to optimum "2. •3. 5. v e l o c i t i e s f o r minimum duct pressure drops, as mentioned e a r l i e r , i s of considerable i n t e r e s t to t h i s t h e s i s . The Pneumatic Conveying of Spheres Through Straight 10 Pipes , by J.A. Hitchcock and C. Jones, contains two em-p i r i c a l c o r r e l a t i o n s between parametric dimensionle.ss groupings of questionable s i g n i f i c a n c e . In scope, they l i m i t themselves to s t r a i g h t pipes, of round c r o s s - s e c t i o n and u n i -d i r e c t i o n a l c o nfiguration and to s p h e r i c a l p a r t i c l e s trans-ported by a i r . However,; two facets of t h e i r work are of i n t e r e s t to t h i s t h e s i s . F i r s t , a photographic stroboscopic technique was used f o r measuring v e l o c i t i e s . This method depends on photographing the number of p a r t i c l e s i n a f i x e d s e c t i o n of duct and r e l a t i n g that number to the i n j e c t i o n r a t e ; i t i s s u i t a b l e only f o r very low feed rates because a c l e a r image of each p a r t i c l e i s imperative to any degree of accuracy. Second, the s l i p r a t i o s determined are very i n t e r e s t -ing and some of them are reproduced here f o r comparison with another writer's work and with the subsequent analysis i n t h i s t h e s i s . TABLE I The S l i p Ratios of Hitchcock and Jones Spherical P a r t i c l e s P p , l b / f t 5 d, i n . VJ , , l b / s e c . m v_, f t / s e c o a S Maple peas 86 . 2 8 .59 71 .58 Maple peas 86 . 2 8 . 2 8 76 .49 Maple peas 86 . 2 8 . 2 8 75 .55 14 Table I (contd). Spherical Particles ? p , l b / f t 5 d,in. ¥ .lb/sec. v ,ft/sec. S Glass halls 155 .28 .31 55 .40 Glass balls 169 .16 .52 75 .42 Glass balls 175 .08 .62 75 .48 Another paper, of particular interest to this thesis, i s Solids Velocities and Pressure Gradients i n a One-Inch Hori-11 zontal Pipe, by J.F. Richardson and M. McLeman . Although this paper does not mention the effects of a i r losses i n the rotary-pocket injector used by the authors, and one might sus-pect these effects were overlooked, the end results, at high vel o c i t i e s , would probably be very l i t t l e affected by a rigor-ous consideration of these losses. Essentially, Richardson and McLeman have shown that the pressure gradient along a straight horizontal duct, during two-phase flow, i s nearly linear. Of course, i t must be remembered that they used f a i r l y low pressures and a f a i r l y short horizontal pipe. But this observa-tion suggests that no violent fluctuations of particle velocity occur due to the presence of particulate material i n the air stream. This inference i s borne out by the consistency of the results obtained by Richardson and McLeman, although their results seem to indicate that the s l i p ratio varies with the a i r velocity, contrary to the results of Hitchcock and Jones. It is worth reproducing here some of the Richardson and McLeman results which relate to s l i p ratios. Particle velocities i n the case of the Richardson and McLeman data were measured by 15 the interrupted flow-pulse technique. TABLE II Slip Ratios of Richardson and McLeman  Particle Shape Diameter,in. Density Free- Range f a l l i n g v a S v & S Range Mean lb/ f t ? ffi°^yft/sec ^ Coal D Rounded .08-dust .04 -87.5 10.7 40 .67 90 .85 Lead Spherical.04-.006 .012 692 26.8 75 .50 115 .66 Aluminum Rounded.015-.004 .009 177 9.9 45 .75 105 .85 A graphical representation of the Richardson and McLeman, and Hitchcock and Jones data i s given in Chapter 4 (Figure 8) , where i t i s considered i n relation to the theoretical considerations presented i n Chapter Several attempts were made by Richardson and McLeman to correlate, empirically, the s l i p ratio with a few conveying characteristics. One such correlation i s linear and suggests that the pressure drop during two-phase flow i s AP = K ^ 0^ m^ —2— where K i s approximately 45000. v & v^ On the other hand, no mention i s made of s l i p ratio variation at very low velocities with particle characteristics, flow charac-t e r i s t i c s , or duct configuration. PRESSURE GRADIENTS R.H. Clark et a l , i n The Pressure Drop During Horizontal 12 Conveyance , consider the effects of the presence of particu-16 l a t e material i n the a i r stream on the turbulence of the a i r and, hence, on the f r i c t i o n a l losses due to the f l u i d alone. I f i t i s assumed that the p a r t i c l e s t r a v e l at a v e l o c i t y lower than the f l u i d v e l o c i t y by exactly the free f a l l i n g v e l o c i t y , t h e i r r e s u l t s are rendered somewhat dubious, which they admit. Obviously, f r i c t i o n a l e f f e c t s on the p a r t i c u l a t e material w i l l a f f e c t the flow of the p a r t i c l e s i n a d i f f e r e n t manner than the p a r t i c l e weight does i n free f a l l . However, t h e i r c o r r e l a t i o n of optimum a i r flow rates with pressure drop i s i n accord with the e a r l i e r mentioned work of Zenz and Rose. Their pressure gradient p l o t s are consistent with the r e s u l t s presented i n t h i s t h e s i s . THERMODYNAMIC EFFECTS 15 . Gas Dynamic Processes Involving Suspended So l i d s , by S.L. Soo, i s the only work reviewed which deals with the thermodynamic processes involved i n two-phase flow. Most writers e i t h e r never mention these e f f e c t s or assume that heat t r a n s f e r between a duct and i t s surroundings i s n e g l i g i b l e , p r i m a r i l y because v e l o c i t i e s are r e l a t i v e l y high. Soo considers not only heat t r a n s f e r between the duct and i t s surroundings, but also between the transporting f l u i d and the in j e c t e d par-t i c u l a t e m a t erial. He notes that a two-phase mixture has f l u i d properties which d i f f e r remarkably from those of a f l u i d alone. Unfortunately, Soo's work i s concerned e n t i r e l y with very f i n e dust (magnesium oxide) where Brownian Movement undoubtedly 17 bears heavily on the results. Moreover, he ignores gravity effects. Conveyability of Materials of Mixed Particle Size, by 14- , P.Ao Zenz , i s concerned with the effects of particle size distribution on flow characteristics. In particular, an empi-r i c a l correlation between the parameters /C D and i s studied with a view to predicting "minimum velocities necessary for saltation-free conveying of mixed size materials under various degrees of solids loading ..." This aspect of pneumatic conveying i s i n i t s infancy and much work remains to be done before we w i l l have a clear understanding of the effects of particle size, size distribution, roughness, shape, density, cohesive properties, electrostatic characteristics, and elastic properties. The literature of the f i e l d , particularly pneumatic convey-ing, i s quite abundant. I ' " ! : . !L/ol ,y ? :ri:-;py •-•'•:-x••> • reviewed; only the better ones have been discussed. The literature varies in approach to the problem from the very general and practical to the empirical and sometimes theoretical PARTICLE CHARACTERISTICS SUMMARY Many more papers than those cited in this chapter were 18 consideration of minutiae. No si n g l e author considers more than a handful of the many va r i a b l e s involved. Nowhere i n the l i t e r a t u r e was there found mention of the co m p r e s s i b i l i t y of a i r and the consequent a c c e l e r a t i o n of both a i r and p a r t i c l e s along a duct. A comprehensive approach i s l a c k i n g i n the l i t e r a t u r e . On the other hand, several authors have made s i g -n i f i c a n t contributions i n the presentation of ideas, observa-t i o n s , empirical c o r r e l a t i o n s , and experimental data. Occasionally throughout the remainder of t h i s t h e s i s , reference w i l l be made to the papers c i t e d i n t h i s chapter, p a r t i c u l a r l y to those concerned with s l i p r a t i o s and the optimization of f l u i d v e l o c i t i e s . CHAPTER 5 0 ANALYSIS A comprehensive theoretical approach to the phenonemon of pneumatic conveying i s lacking i n the literature; this has already been pointed out. One of the few works to date which purports to be theoretical i s the paper by Vogt and White. They tackled the problem from the standpoint of energy consider-ations f o r the s o l i d phase and the basic work-energy equation for the f l u i d . However, many of the variables which confound the problem render this somewhat general approach inadequate. Therefore, an attempt w i l l be made herein to consider the aero-dynamic and f r i c t i o n forces involved i n two-phase flow. For purposes of generality, two-phase flow w i l l be considered where a compressible f l u i d travels i n a duct of any cross-sectional shape, any geometric configuration consisting of straight sections inclined at any angle with the horizontal and curved sections i n any plane, there being particles of any shape, size, roughness, sij;e distribution, density, electrostatic character-i s t i c s , and e l a s t i c properties carried along the same duct at some relative velocity by the compressible f l u i d , the duct being situated i n any environment. DESCRIPTION OF FLOW The description of two-phase flow i n pneumatic conveying, given by Korn, has considerable merit. That i s , some evidence 20 does e x i s t which i n d i c a t e s t h a t a v e l o c i t y g r a d i e n t from the o u t s i d e to the c e n t e r of a duct c r e a t e s l i f t f o r c e s which tend t o draw p a r t i c l e s toward the c e n t e r of a duct. F i g u r e 3(a) i l l u s t r a t e s Korn's concept, as he i l l u s t r a t e d i t . Note t h a t although Korn's s t r e a m l i n e s are not i n accord w i t h c o n v e n t i o n , at l e a s t they suggest what occurs i n the f l u i d s urrounding an i n d i v i d u a l p a r t i c l e . F i g u r e 3(h) i s probably a b e t t e r i n d i c a t i o n of what a c t u a l l y occurs to an i n d i v i d u a l p a r t i c l e . The l i f t f o r c e i s p r i m a r i l y due to the Magnus e f f e c t which accompanies the r o t a t i o n a c q u i r e d by the p a r t i c l e through f r i c t i o n a l c o n t a c t w i t h the duct. Beyond t h i s d i s c u s s i o n , however, i t i s not considered worth s p e c u l a t i n g here on what a c t u a l l y occurs i n the t u r b u l e n t f l o w around a randomly moving, i r r e g u l a r l y shaped p a r t i c l e . Although such a tumbling p a r t i c l e no doubt has many of the a t t r i b u t e s of a sphere, such concepts as c i r c u l a t i o n seem to have l i t t l e r e l e v a n c e to t h i s d i s c u s s i o n . Moreover, the i n t e r -a c t i o n of p a r t i c l e s w i t h one another and w i t h the duct w a l l s renders the c o n s i d e r a t i o n of i n d i v i d u a l p a r t i c l e s a l i t t l e f u t i l e . Consequently, i n s o f a r as t h i s a n a l y s i s i s concerned, l i f t f o r c e s on the p a r t i c l e s w i l l not be considered d i r e c t l y but w i l l be taken i n t o account i n d i r e c t l y i n a general f r i c t i o n f a c t o r . The f l o w of a s o l i d - f l u i d mixture w i l l h e r e i n be considered 21 as the flow of two separate f l u i d s . An analogy w i l l he drawn between the random motion of particles, and their relationship to the duct walls, and the random motion of molecules which gives rise to the hypothetical f l u i d f r i c t i o n which i s so basic to the study of single-phase f l u i d flow. LIFT i LIFT 3(a) 3(b) Korn's Description A More Likely Explanation Fig.3 L i f t Forces on a Single Particle BASIC EQUATIONS 16 The general flow equation for a compressible f l u i d , where the acceleration of the f l u i d i s negligible and the Mach number i s less than 0.188, reduces to the Darcy-Weisbach equation, dP f „ Q v c ax = 1 \ax ax dx 2Dg 3 ( D where f^ i s dimensionless and D i s a linear dimension associated 22 with the perimeter of the duct. For the flow of the two-phase mixture i n the duct shown in Figure 4, the pressure drop due to the f l u i d flowing i n the duct i s - dP f 1 ? a x v a x 2 d x 3(2) ax " 2Dg The macroscopic random motion of the particulate phase i s perhaps analogous to the microscopic molecular motion of the f l u i d phase and, consequently, should be measurable indirectly through measurement of the f l u i d pressure and temperature. Hence, due to the random motion of the particles, contact with the duct walls, and contact with the a i r molecules, there i s a further pressure drop due to the passage of the particles over the distance dx. This material pressure drop might be expressed i n the same terms as the f l u i d pressure drop as shown i n 3(2), where f ^ i s a f r i c t i o n factor equivalent to f^. *,P v 2 '4^-mx mx f  2dx •^p tf>- vnv TT1Y mx 2Dg But P i s merely m^ . Therefore, V m x _ dP _ 4 m mx ^mx 2DADg ^ > Equation 3(2) determines the pressure drop due to the passage of f l u i d relative to the duct wall; 3(3) i s the equivalent equation 23 for random motion of the particles. How i t i s necessary to consider the pressure drop due to the f l u i d passing through the constriction created by the parti c l e s . To do this we must remember that the f l u i d and the particles have a relative velocity u . Prom the standpoint of the f l u i d then, the particles might be considered to be fixed in space. The f l u i d must pass around and among the particles. This flow and the Fig.4 Two-phase Flow i n a Straight Duct at some Point a Distance x from the Particle Injection Point. pressure drop i t produces should be ascertainable with reference to the Darcy-Weisbach equation and the concept of the "hydraulic radius". In the immediate v i c i n i t y of x, the particle distribution along the f i n i t e duct length w i l l be f a i r l y uniform, although 24 the concentration across the duct w i l l depend to some extent on the o r i e n t a t i o n of the duct and on whether the duct i s curved or s t r a i g h t . Consider the c o l l e c t i o n of p a r t i c l e s i n the duct length to be represented by "standard" or "average" p a r t i c l e s of mean weight w, mean surface area S^, mean cross-s e c t i o n a l area A , mean volume V , and s o l i d density P . Let the number of p a r t i c l e s i n length AL be N. Since the density of the p a r t i c l e d i s t r i b u t i o n i s P = ^ m at x, the A Dv number of p a r t i c l e s i n length AL i s N = V m A L 3(4) w v ^ The t o t a l p a r t i c l e surface area over which the f l u i d must pass as i t t r a v e l s AL i s WAL S^ Area - * P. 3(5) mx . The wetted perimeter due to the p a r t i c l e s at x i s then Perimeter = m p 3(6) w v „ mx The flow area between the p a r t i c l e s and the duct walls can be determined on the same b a s i s . I t i s merely the pipe cross-s e c t i o n a l area l e s s the area obstructed by the particles« The area obstructed by the p a r t i c l e s i s A ' = ^ ! E 5(7) mx 25 The flow area then becomes A f = A D ~ A p w v mx Now, by d e f i n i t i o n , the hydraulic radius f o r any flow c o n f i g -u r a t i o n i s R _ x-s e c t i o n a l flow area H wetted perimeter Therefore, at x, A D " Wm V p w v f f l T A , V %x° - r F - V 5(8) 1£5 p p w v m v mx Introducing another f l u i d f r i c t i o n f a c t o r , f 2 , we have, as a pressure drop over dx f o r the passage of the f l u i d over and among the p a r t i c l e s , - d P „ _ = f 2 ?a^Jd* = f??ax u x 2 ta 5(9) Therefore the t o t a l pressure drop f o r the. f l u i d and p a r t i c l e mixture passing over the distance dx i s ; dP = dP + dP + dP x ax mx amx - = f 1?ax y 2 a x + h Vm vmx + f2 ?ax ^ x 3(10) dx Or, w r i t t e n another way, 2 l^ax v ax + 2 D g 2 D A D g 2 S (" V i T^Q \ % V The presence of the r e l a t i v e v e l o c i t y term, u , i n equation 3(10), makes the equation d i f f i c u l t to deal with. However, by 26 consideration of p a r t i c l e motion, other than random motion, and p a r t i c l e motivation, we can eliminate the u x term. Consider the mean p a r t i c l e i n Figure 5« The drag forces on the mean p a r t i c l e overcome f r i c t i o n a l forces, as accounted f o r by F, and accelerate the p a r t i c l e . Therefore 2 w d 2x + F = CD Ap?ax u x 3(11) g d t 2 ° r u 2 2 R < ^ m x d > + F ] 3(12) x " CD Ap Pax ' S ^ ^ Combining 3(10) and 3(12) we have: ^ m x d vmx + F > ~ ^ x " f 1 <?axv ax + W m x + W m ( S > 3(13) o l T 2gD HgDAp C p A ^ w v ^ 1 - V pV m ~ Equation 3(13) i s general and should apply to f l u i d i z a t i o n and hydraulic conveying as we l l as to pneumatic conveying. However, consideration of i t s a p p l i c a t i o n to these two r e l a t e d f i e l d s i s beyond the scope of t h i s thesis and w i l l be l e f t to future researchers. The a p p l i c a t i o n of equation 3(13) to pneumatic conveying i s made d i f f i c u l t by the co m p r e s s i b i l i t y of a i r and, hence the dependence of £ > , v_ . v_ . and F on pressure which, i t s e l f , i s dependent on x. Presumably a set of functions might be found to describe these v a r i a b l e s i n terms of x and thereby permit the numerical i n t e g r a t i o n of 3(13)• However, i t i s considered here that numerical methods are extremely l i m i t e d 27 i n t h e i r usefulness and therefore should he considered only as a l a s t r e s o r t . Consequently, i t i s considered more appropriate to t r y to r e v i s e the theory and perhaps s i m p l i f y the basic equa-t i o n through the use of a l i t t l e imagination, such that the f i n a l r e s u l t i s v a l i d , sensible, and as u s e f u l as possible over as wide a range of a p p l i c a t i o n s as p o s s i b l e . Fortunately, the case of pneumatic conveying permits such s i m p l i f i c a t i o n . The various steps i n t h i s s i m p l i f i c a t i o n w i l l now be considered as a prelude to the i n t e g r a t i o n of equation 3(13)« SIMPLIFICATIONS FOR PNEUMATIC CONVEYING The E l i m i n a t i o n of V p Vm i n Equation 3(13) w vmx AD Considering most a p p l i c a t i o n s , and p a r t i c u l a r l y the experi-mental system used i n t h i s research, the various terms i n V W p m can be l i m i t e d as follows % w v^A-n mx D ^ _w = O 50 l b / f t 5 V P P V < 48 l b / s e c . - f t 2  vmx > 4-0 f t / s e c . Therefore V p Wm / 48 = .024, or approximately 2#%. This w Vmx AD term expresses the volume occupied by the p a r t i c u l a t e phase of the two-phase flow i n pneumatic conveying; i t seems reasonable to drop t h i s term f o r s i m p l i f i c a t i o n purposes. Note, however, that such s i m p l i f i c a t i o n i n hydraulic conveying and f l u i d i z a t i o n i s not possible because v e l o c i t i e s i n hydraulic conveying are 2 8 approximately one-tenth those required i n pneumatic conveying and, i n f l u i d i z a t i o n , t h i s term becomes @ m/£p» which v a r i e s from 0.4 to 1 .0 f o r most p a r t i c u l a t e materials. Determination of Pressure as a Function of x 12 11 I t was shown by Clark , Richardson , and several others, and i n f a c t i t w i l l be shown again i n the experimental apparatus used i n t h i s research, that the pressure gradient i s e s s e n t i a l l y l i n e a r , f o r high v e l o c i t i e s and small pressure drops, over an entire system, regardless of geometric configuration. Of course, small f l u c t u a t i o n s occur at elbows, but these f l u c t u a t i o n s are purely l o c a l . Figure 2 2 , i n Chapter 7* shows t h i s l i n e a r char-a c t e r i s t i c . I t i s therefore reasonable to e s t a b l i s h an approxi-mate l i n e a r function to describe pressure along the duct i n terms of x. AP = P n - P 2 3(19) P v = P. - AP x L L i s the equivalent length of the duct. Equivalent lengths merely account f o r added f l u i d presstire losses at elbows; they may be c a l c u l a t e d by various empirical methods av a i l a b l e i n any handbook on f l u i d mechanics. For the purposes of the t h e s i s , the equivalent length was determined by measuring the t o t a l pressure drop over the system and r e l a t i n g that drop to the i n -d i v i d u a l drops over several of the s t r a i g h t sections i n the system. 29 Use of Basic Gas Laws I t i s known that flow i n an uninsulated duct at very low v e l o c i t i e s i s e s s e n t i a l l y isothermal. Flow through a nozzle on a short insulated pipe can approach an adiabatic process. Pneumatic conveying, however, generally e n t a i l s the use of a c o n t i n u a l l y operating compressor or blower which supplies a i r at a temperature considerably above that of the duct e x t e r i o r . Consequently a small amount of heat t r a n s f e r occurs out of the system and t h i s , combined with i n t e r n a l f r i c t i o n a l e f f e c t s , r e s u l t s i n a process which i s neither isothermal nor adiabatic. Fortunately, the basic gas law PV n = constant 3(20) s t i l l a p p l i e s , although the p r e d i c t i o n of the value of n i s often d i f f i c u l t . I f we proceed, using a generalized n, i t should be possible to e s t a b l i s h l a t e r a c r i t e r i o n upon which to base i t s determination. In any case, the e f f e c t of n i s small and a con-servative value can always be used i n the absence of a more accurate or predictable value. Equation 3(20) permits us to describe the gas conditions at x i n terms of those at the i n i t i a l or i n j e c t i o n point of the system. The i n i t i a l conditions are what u l t i m a t e l y must be determined i n the design of a system. P x V l l x = P 1 v I l 1 3(21) I f the duct has a constant c r o s s - s e c t i o n , then ) 0 va1 £ ^P x 30 Prom c o n t i n u i t y considerations, that i s , since the f l u i d flow i s steady, ?ax v a x - 9 a l va1 5 ( 2 5 ) I t therefore follows that i n the i n t e g r a t i o n of the f l u i d pressure l o s s term i n equation 3(13)» f1 ?ax vax = f1 ?a1va1 ( f j . ) 2 g D 2 g D ( P x ) Now, using 3(19), we have f1 Tax vax = f 1 Pa1va1 (1- AP X ) 3(24) 2 g D 2 g D LP 1 3 1 1 ( 1 ( )-1/n v a x - va1 < 1 - 3(25) ^ a l ( LP 1 ) Equation 3(25) describes the f l u i d v e l o c i t y at any point along a duct i n terms of the distance from the i n i t i a l or i n j e c t i o n point i n the system to the point under consideration. The con-cept of the v e l o c i t y s l i p r a t i o ( S - vmx^ vax) permits the use of equation 3(25) i n the determination of p a r t i c l e or material v e l o c i t y at any point i n the duct. Hence, we have ( )-1/n vmx " 3 v a x - S va1 ? ' §f ] 3(26) I t w i l l he shown presently that the s l i p r a t i o , although having a constant rate of change with respect to a i r v e l o c i t y at high a i r v e l o c i t i e s , i s not i t s e l f constant except under s p e c i a l circumstances. 31 Calculation of F g for Straight Ducts Consider a single particle passing along a duct. The resistance which aerodynamic forces must overcome consists of f r i c t i o n a l and gravitational forces. Part of this resistance w i l l he made up of sliding f r i c t i o n between duct wall and particles and the gravitational attraction of the earth; the other part w i l l be due to the random motion of the particles, as discussed under BASIC EQUATIONS. Consider the mean particle i n Figure 3>» The resistance offered the particle due to random motion, f r i c t i o n a l contact with the duct, and elevation change i s where F i s the resistance force, due to random motion, acting on one particle i n duct length With reference to equations 3(3) and 3(4) and noting that N i s approximately constant over A L , i t i s apparent that F may be expressed as follows: F Hence, 2 2 F F i n a l l y , 3(27) 32 Calculation of F for Ducts with Constant Radius of Curvature Consider the curved duct, with radius r to i t s centre-l i n e , lying i n the 1' -2' plane i n Figure 6. «>< i s the angle between the horizontal 1-2 plane and the 1'-2' plane. 9 i s the angle the radius vector makes with the horizontal 1' axis, measured i n the 1'-2' plane. ^ i s the angle between the 1 and the 1' axis measured in the horizontal or 1-2 plane. It i s clear that appropriate values of w?, 9 and v^ ) may be found to Fig.5 Resistance offered to a Single Particle describe any arc on the surface of a sphere. Consequently, an arc or a curved duct having any geometric orientation may be described by appropriate values ofc*, 9 and <p . But, since gravity acts v e r t i c a l l y downward along the 3-axis, since i t i s the only external force influencing flow i n a curved duct, and since the 3-axis i s normal to the plane i n which ^ i s measured, \p> i s arbitrary and, for the purposes of these considerations, may be taken to be zero. 33 A simple transformation from the unprimed coordinate system to the primed system greatly f a c i l i t a t e s analysis of the forces involved i n flow through a curved duct. The gravitational force on a mean particle of weight w i s a tensor of rank 1. Its representation i n the unprimed coordinate system i s G = J i x G Fig „ 6 Curved Duct i n Cartesian Coordinates Now, G1 = G 2 = 0 and -w. Therefore 2 G' = =w sino<> G 1 ^ a - W COS&C In vector notation, - ( - r ' - ) G = w (- 0 sine* •- k' coswj 34 The c e n t r i f u g a l force may he designated by C = Cr = Cu„ m where _ _ T = r ( I* cos © + j ' s i n 9) u r= T'cos 9 + " J ' s i 1 1 © U Q = - 1'sin 9 + j 1 cos 9 Assuming the material v e l o c i t y i s constant, or that an average value may be used, 2 - 2 w v^ u w v„ C = g ^ r = - g j r - d ' c o s 9 + J'slii 9 ) The t o t a l force perpendicular to u 9 may be determined i n magnitude by taking the cross-product of u 9 and (G + C"). p p | - |(G + C) x u Q| _ _ ' ' / i v 2 2 „i \ G + C = w >i m cos 9 + j ( m s i n 9 - sino.)-k cose* ^ gr gr ; / 4— ; 2 ~ [ N J = w y 1 + vm - 2 vm sin<=*sin 9 - cos 9 s i n * * ' g 2 r 2 s r The tangential force due to G + C i s defined by |f F| = - ( G > C). u 6 w s i n ©< cos 9 As i n the case of s t r a i g h t pipes, the resistance due to i n t e r n a l f r i c t i o n i s 2 "2gD" 35 Hence the total resisting force i n a curved pipe i s F C f 3 i53»i + w + I T F I = w ( f,\ 1 + vm - 2 vm sino<,sin 9 - cos 2© sin2e*. g2r2 S r 2 ) + sino<cos 9 + 14 vm ) 3(28) 2gD ) In the lim i t where r o o and 9 #-0, ( - v 2 ) F„ = w (f,cos«x.+ sin«K+ x4 m ) = F„ C ( 5 ~2gD~ ) 3 F reduces to F- i n the l i m i t , thereby behaving sensibly and as would be expected. Now, equation 3(28) may be re-arranged such that • .  wf,v3 S \L . 2 2 ~ • _,_ r, _2_2_ 2rt_,_2 F Q = 3 m ( |/1 + g r = 2gr s i n x sin 9 - g r cos 6sinB< . . « * . < • v - v m m f _ ) + K r sino<cos 9 + 4 ) 3(29) Since equation 3(29) must ultimately be integrated with respect to 9, i t s simplification i s imperative. Several simpli-fications seem readily apparent. For instance, for large values of v m and small values of r, taking into account the fact that f ^ ^ f j , as w i l l be shown presently, F c becomes Fc = 5vm 3(30) gr If equation 3(29) cannot be simplified as shown i n 3(30), 36 then at least the use of average values of sin9, cos9, and 2 cos 9 w i l l release F Q from i t s dependence on 9. Hence, for an arc of 9 radians between and 9 2, r ©p cos9x,-cos9o sin9d9 = ' Q d 3(31) 2 1 sin av 92~©x| cos 9 1 av ~ 92-9/j -4 1 Q / 2 sin9p=-sin9x, cos9d9 = Q Q 3(32) 9 2 " @ 1 7 @ 1 r 9 2 c o s 2 e a v = 9 2^9 1 c os 29d~ = 1 (1 + s l n 0 2 e o B G 1 - a l n 0 1 o o s 0 2 ) 3 ( 3 5 ) 9p ~ 9,. Slip Ratios for Any Duct Before i t w i l l be possible to use the s l i p ratio concept i n conjunction with equation 3(26) for determining the material velocity at any point i n a duct, i t w i l l be necessary to c a l -culate i t separately for straight and curved ducts i n terms of the appropriate resisting forces F g and F Q o Once the s l i p ratios for F and F are known, i t w i l l be possible to determine the average s l i p ratio for a duct of any geometric configuration. The work of Richardson and McLeman, and Hitchcock and Jones, indicates that the s l i p ratio may be constant or may vary s l i g h t l y with a i r velocity; i n either case, gradually acceler-ating flow or non-accelerating flow may be adequately covered by assuming the s l i p ratio to be independent of the air velocity 37 other than by d e f i n i t i o n . This consideration obviously ignores the unstable condition i n the immediate v i c i n i t y of the i n j e c t o r where the material receives i t s i n i t i a l a c c e l e r a t i o n . Moreover, i t i s necessary to consider the obvious f a c t that the s l i p r a t i o v a r i e s , depending upon the i n c l i n a t i o n of a s t r a i g h t duct, and upon the curvature and o r i e n t a t i o n of a curved duct. Consideration of these s l i p r a t i o s i s further f a c i l i t a t e d by the assumption that a sudden change i n duct c h a r a c t e r i s t i c s , f o r instance, from a h o r i z o n t a l duct into an elbow of any o r i e n t a t i o n w i l l cause an appropriate change of the s l i p r a t i o . This w i l l be true i n e f f e c t even though the changes w i l l un-doubtedly occur over a f i n i t e period and the various s l i p r a t i o s w i l l l a g , i n propagation, with respect to the p h y s i c a l o r i e n t a t i o n of the duct. The s l i p r a t i o f o r non-accelerating flow i n any s t r a i g h t duct i s obtained from equations 3(11) and 3(27)« Hence, w ( F 5 c o s « * + sino< + f W 2 ) = V P gax ( v a x ~ vmx ^ ^ 2gD J 2g By d e f i n i t i o n , f o r a s t r a i g h t duct and non-accelerating flow, S = - S s v a Therefore, •f,cosK- + sinc<+ V s 2 v a 2 = CD A p ? a <1 " S s ^ 2gD 2gw 38 Solving the above quadratic equation for S we have 3(34) '1 " ' 1 " W a D 3(35) 2wg (f-,cosc<+ sin©< ) B o = 1 ™ ^ o 5(56) CD Ap ?«V The negative root i s taken i n equation 3(34) because f^ and f ^ must both be positive and S must clearly be less than one i f s the particles are to be motivated by the a i r . The determination of the s l i p r a t i o i n a curved duct of any t o t a l angle 9 and any radius of curvature r i s complicated immensely by the mixed nature of the value of F_ given i n equation (3(29). If a constant material velocity v m i s assumed, then, using equations 3(11) and 3(29) and carrying out arithmetic manipulations similar to those in the case of the straight duct above, sixuKcos© + £ 4 r ) 3(37) The use of equations 3(31)» 3(32) and 3(33) for the average 39 p values of sin© , cos©, and cos © i n the case of a curved duct of t o t a l angle © and constant radius of curvature r , together with an i t e r a t i v e technique such as that used i n t h i s t h e s i s (see Appendix I I ) , w i l l f a c i l i t a t e the s o l u t i o n of equation 3(37) f o r S . Unfortunately, i n most pneumatic conveying a p p l i c a t i o n s , none of the terms on the righthand side of equation 3(37) may be dropped, i f a high degree of accuracy i s to be obtained. Bearing i n mind the current lack of knowledge regarding the drag c h a r a c t e r i s t i c s of i r r e g u l a r l y shaped, tumbling p a r t i c l e s and the i m p r a c t i c a b i l i t y of evaluating p a r t i c l e properties s t a t i s t i c a l l y , i t i s well worth noting that the lefthand side of equation 3(37) may be s i m p l i f i e d by considering the r e l a t i o n s h i p between p a r t i c l e c h a r a c t e r i s t i c s and the f r e e - f a l l i n g or terminal v e l o c i t y , v f , of p a r t i c l e s , e i t h e r s i n g l y or i n groups, i n a i r whose density i s the same or almost the same as that flowing i n the duct. Such a considera-t i o n w i l l be made l a t e r i n Chapter 4 i n connection with the work of Richardson and McLeman, and Hitchcock and Jones„ F i n a l l y , once the various s l i p r a t i o s f o r a l l of the com-ponent duct sections i n a complex geometric configuration have been c a l c u l a t e d through the use of equations 3(34) and 3 ( 37 ) , i t i s possible to c a l c u l a t e the average s l i p r a t i o f o r the duct configuration. This may be done by noting that the t o t a l "residence" or "duration" time f o r the p a r t i c l e s i n the duct configuration w i l l be the sum of the duration periods the p a r t i c l e s spend i n each segment of the duct. 40 That i s , t^. — t/q + + t j + t ^ + o o o t ^ where t^. i s the t o t a l time and t^ , t 2 * t ^ , .». t n are the duration periods of the p a r t i c l e s i n n d i s t i n c t s t r a i g h t and curved duct sections. Then, noting that t ^ = L / v m a v , t^ = l"\/vmss\ s and so f o r t h , the average s l i p r a t i o f o r a complex geometric duct configuration c o n s i s t i n g of p s t r a i g h t ducts and q curved ducts may be found by considering the f a c t that L = 1 + 2 + .. p + \ 1 + 2 2 + ... q q  vmav vms1 vms2 vmsp Ymc1 vmc2 vmc P q L or g S a v and, S a T - H s 5(58) xm+1 ~ xm Ssm S e l e c t i o n of a Function to Describe C D and f^ i n Terms of Reynolds Number L i t t l e i s known about the drag c o e f f i c i e n t s of tumbling, i r r e g u l a r l y shaped p a r t i c l e s . Moreover, groups of such p a r t -i c l e s of a broad s i z e d i s t r i b u t i o n may well have drag charact-e r i s t i c s which are unrelated to those of si n g l e p a r t i c l e s . However, since the work i n t h i s t h e s i s i s p a r t i a l l y concerned with p a r t i c l e s of uniform si z e and shape, and since much of the work done to date by others i s concerned with spherical p a r t i c l e s , consideration i s now given to the drag c o e f f i c i e n t s of spheres. 1 5 I t was shown by Stokes x that f o r a sphere moving i n a f l u i d , and having a Reynolds number l e s s than one, the drag force i s simply I t i s known, as w e l l , that f o r very high v e l o c i t i e s , where the Mach number approaches 0 .7? the drag c o e f f i c i e n t becomes f a r le s s dependent on the Reynolds number than on the Mach number. However, i n pneumatic conveying, a i r v e l o c i t i e s and p a r t i c l e c h a r a c t e r i s t i c s are such that the drag c o e f f i c i e n t of a sphere i s p r i m a r i l y dependent on Reynolds number, whose value ranges from 10 to 1 o \ Several sources e x i s t where information on drag c o e f f i c i e n t s f o r spheres i n t h i s range of Reynolds number may be 1 5 obtained. One such source i s F l u i d Mechanicsr y by Streeter. Unfortunately, a su i t a b l e function describing C-^ i n terms of R e p over the range of R from 10 to 10 i s not given by Streeter. Consequently, f o r convenience i n the analysis of the experimental r e s u l t s , both manually and with the aid of the computer, a s u i t -able f u n c t i o n was sought. Examination of Figure 4—23 i n Streeter's text suggested that the function F - 5 TT d/< u ,n ep 3(39) 42 where a , h, m, and n are constants, would f i t the data ade-quately. Four points on the curve given by Streeter were selected such that Rep1 = 10 °D1 « 4.00 Rep2 = 10 2 CD2 = 1.00 R e p 3 = 10 5 °D3 = 0.42 Rep4~~ 10 4 . °D4 = 0 .39 Since the various values of R were selected so as to be ep related i n powers, the constants in 3(39) were found to be n = log In °D4 - 3 In GD2 ^D1 5)1 - 2 In 2l>l - 2 In CD2 °D1 °D1 3(40) b = m = a = In CD / CD1 \ In CD2/Gd1  R l lep| R n ep3 " 3 R ne-P3 + 2 In °D5 / CD1 " b.. ^ e p l ^ e p l 4.0605 'D2 - 1 -.m 5n 3(41) 3(42) 3(43) R ep2 <2> R ep2 Equations 3(40), 3(4-1), 3(42) and 3(43) provide the values which give equation 3(39) the form, for a sphere, of 1 016 C D = 7»00 R~p* e(1 . 1 5 R e p 1 8 7 5 > 3 ( 4 / ° Equation 3(44) precisely f i t s the data given by Streeter for A C C U R A C Y OF F O R C E - F I T FUNCTION -I.OI6 U I S O R p t > ^ C D = 700 Rep P • F O R ;AI_< F .A IT TI f >C > 1 S 0 1 N T S c F 2 M Ro x k :u >* A n IE \E •t > E. C U R V E >:PERIM D E ' =MT MA re. @ / f • \ \ \ EX PE 311 It n A 1. CURV E F .^Tl * o =T CP \ \ Fig.7 Accuracy of the Fo r c e - F i t Function f o r C~. 4* 44 10<R <10 , as shown i n Figure 7° Hence, this function i s — ep— satisfactory for the purposes of this thesis. Coincidentally, the same function f i t s the data presented i n Flow of F l u i d s ^ , by Crane, for the f r i c t i o n factor f^ with respect to compressible flow i n clean steel c y l i n d r i c a l ducts. • . ^ - 0.01084 B0.03067 ^21.8 R-0.3586 5 ( 4 5 ) This equation precisely f i t s the Crane data for Reynolds 3 7 numbers ranging from 4 x 10 to 2 x 10'. Pneumatic conveying normally involves values of Reynolds numbers ranging from 5 x 10^ to 5 x 10^; hence, again, the function given i n 3(39) is satisfactory for the work i n this thesis and i n the general study of pneumatic conveying. INTEGRATION OF EQUATION 3(13) FOR PNEUMATIC CONVEYING The general form of the equation for pressure drop during two-phase flow was earlier shown to be -dPx = f1?ax vax 2+ f4Wmvmx + f 2 Sp Wm [R ~d>P t ? ) ox°* 2gD ^517- O ^ W T ^ ( 1 - V p V •) w mx^ D This equation may now be simplified and integrated by the following steps; (a) Drop the term p^^ m . It has been shown that this term wv A D may be dropped for pneumatic conveying without incurring significant error. 45 Use equations 5(19), 3(25), 5(25) and 5(26) to express air density, a i r velocity, and material velocity i n terms of x. Multiply through by dx and rewrite equation 5(15) as - 1/n flPalV) 2 + W av val ] ( 1 - ^ I <*x 2gD 2gDAD ) ( LP 1 ) ( ( ) V n ) f2 wm Sp (w dvmx + F (1 - APx ) dx ) 3(46a) CDApADw(I v a 1 S ( LP 1 ) ) Group the terms independent of x i n equation 3(46a) and designate the groups as ' f1?a1 va1 2 + f4 Wm Sav va1 = —2"p + 2gDAD B = 2 M P C A A Hence, equation 5(4-6a) can he rewritten as -1/n ( 1/n ) . (1 - APx ) dx + B (w dv m V + F (1- APx) dx )5(46b) LP~ w" (g v ^ 3 LF/| ^ Integrate the lefthand side of equation 5(46b) between the i n i t i a l pressure, , and the f i n a l pressure, P^, such that -dP = P 1 - P f = AP P1 46 Integrate the f i r s t term on the righthand side of equation 3(46b) over the equivalent length of the duct, expand the resulting expression, and drop fourth and higher order terms, such that L C - V n _ A ( 1 - £Px ) dx " A L (1 + AP ) 0 Integrate the velocity term on the righthand side of equation 3(46b) from zero at the injection point to the f i n a l velocity, which depends on the s l i p factor i n the f i n a l section of duct, Jmf B dv _ = B v » mx — mf S Integrate the last part of equation 3(46b)(that part involving F) over each of the straight and curved sections in the duct configuration independently, using the different values of F and S given i n equa-tions 3(27), 3(29), 3(34), 3(37) and 3(38), but noting that the values of F and S i n a given section of duct are independent of x and dependent only upon the fixed or constant duct orientation in space and whether the duct i s curved or straight„ Upon integration, expand 47 the resulting expression and drop fourth and higher order terms. Consequently, for a given geometric configuration having p straight ducts with various inclinations with respect to the horizontal, appro-priate s l i p ratios, and each section being defined by ^ xm+1 ^asured along the duct from the injection point, the last part of equation 3(46b) becomes xm+1 Is / (1-APx) dx S g I LP, V (r 3coso. m + s i n c ^ ^ x ^ ^ -x m )_ p ( x m ^ - x 2 ^ v a i ; s s m ( S E E f ) B v a 1 f A Ssm ((xm+1~ xm) + A P _ (^m+l " *^ m) ) 2gD / ( 2UP^L ) m = 1 For a given geometrical configuration having q curved ducts lying i n planes with various inclinations with respect to the horizontal, a constant radius of curva-ture r , each having total angle © m, and each having i t s in l e t point defined by x m measured along the duct from the injection point, the last part of equation 3(46b) 48 becomes q xm+1 -1/n B \ F c (1 - APx ) v a i w Z ^ L P 1 m = 1 JL- ( ) Bv /i \ S._ ( f3 C + f4 rj(Q + AP (2x_Q_+ r© 2 )) a1 \ cm ^  2gB C m 2l3yL m m m ) m = 1 * - ) - p ' + B \ Cr s i n g m c o s e a v ) ( 9 m " (2xm9m + r 9 m> m = 1 where xm+1 - xm = r © and the involved term in the P expression for c (equation 3(29)) i s , for convenience, given by C. That i s , G + -f i f jE ~ 2p:r sin«xsin©av - g 2r 2cos 2© a vsin 2 s < v v m m (h) A l l of the 4 terms i n each of the above expressions for p straight and q curved duct sections may be com-bined to simplify the ultimate expression. That i s to say, q B v a 1 f 4 r V""Scm£em + AP (2 x © + r© 2 )t 2g5— ) I 2 ^ L " m m m ] m = 1 J r ^ 2 2 ) + B v a 1 f 4 V Ssm ((xm+1-xm)+ AP ( xm+1- xm))= B f4 va1 Sav L(1+ AP ) 2gD ) ( 2nP1L ) 2Dg 2nP^ m = 1 4-9 ( i ) F i n a l l y , then, equation 3(4-5) becomes AP = AL (1+ AP)+ B [Sfva1 (1+ AP)+f4-va1 S a v L ( 1 + AP) ] 2nP^ ( g nPx, 2gD 2nPT[ ) P m = 1 T V 5 0 0 S , V s i n o m ) [ ( V l - x i ) - A P (x2m-f1-x2m)] V l S s m L 2 ^ J ( Scm f3 va1 C + r s i n " m c ° 3 9 a v ) j V ^ C 2 ^ V r Q m > ] S va1^cm 3(50) where A, B and C are as previously defined. Equation 3(50) provides a means whereby the pressure drop across the ends of a duct containing two-phase flow and having any geometric configuration may be calculated by an iterative process. SUMMARY It has been shown in this chapter that a theoretical approach to two-phase flow leads to a general equation which describes the three basic two-phase flow processes, namely, f l u i d i z a t i o n , hydraulic conveying, and pneumatic conveying. Application of the general equation to pneumatic conveying was treated in rigorous d e t a i l . The f i n a l outcome was a basic equation which describes two-phase flow i n a closed duct of any geometric configuration, where onejphase i s a compressible 50 f l u i d and the other i s a particulate s o l i d . In addition, a l l of the variables which enter into the design of a pneumatic conveying system, such as duct diameter, f l u i d velocity and density, and particle shape and size are taken into account either d i r e c t l y by their presence in the f i n a l equation or indirectly through the determination of such quantities as the s l i p ratio and various f r i c t i o n factors. Consequently, an equation has been derived whereby the pressure drop along a duct, i n which a two-phase mixture flows, may be determined and, as w i l l be shown in the next chapter, an optimum f l u i d velocity may be determined for minimal pressure drop. CHAPTER 4 COMPARISON OF THEORY WITH THE WORK OF OTHERS The a n a l y t i c a l development presented i n the previous chapter might well be viewed now from the standpoint of other research. I t was noted i n the Review of the L i t e r a t u r e that the work of Zenz, Rose et a l , Richardson and McLeman, and Hitchcock and Jones was of considerable i n t e r e s t to the work i n t h i s t h e s i s . The a p p l i c a t i o n of t h i s other research w i l l now be considered with reference to s l i p r a t i o s and the optimiza-t i o n of flow v e l o c i t i e s f o r minimal duct pressure drop. CONSIDERATION OF SLIP RATIOS The work of Richardson and McLeman, and Clark et a l , has suggested that the s l i p r a t i o f o r a given material may vary with a i r v e l o c i t y . The r e s u l t s presented i n both papers re s u l t e d from work done with h o r i z o n t a l pipes, f o r which equations 3(11) and 3(27) become w d 2x + w ( f 3 + V m x J = C D A p ? a x u x 2 4(1) g ^ 2 ( 2Dg ) 2g Now, because the above mentioned work was conducted i n short sections of duct, which were considerably removed from the i n j e c t i o n point of the p a r t i c u l a t e m aterial, and since both writers neglect the a c c e l e r a t i o n of both the gaseous and 52 particulate phases, we may assume that d 2x ^ 0 I? hence, equation 4(1) becomes f 5 + V m 2 = V l ^ a ( va " vm) 2 4(2) 2Dg 2gw Note that Richardson and McLeman determined the free f a l l i n g velocities for their materials; see Table I I. But for a free-f a l l i n g particle descending at i t s terminal velocity, w = W a v / 2g For spherical particles w = . T T * \ 6 and Therefore 4 CD Ap?a = 2 = 1 4(3) 2 S W 4 gd^T ^ 5 Equation 4(3) makes i t possible to rewrite equation 4(2) such that the particle properties are taken into account by the free-f a l l i n g velocity of a single p a r t i c l e . Hence, V + Vaf = £ Va - vm ) 2 4(4) 2Dg > v ( a - m. j ( T f j 53 The dependence of vm on v a and the other variables in equation 4(4) suggests that the evaluation of vm would be greatly f a c i l i -tated by rewriting equation 4(4) as follows: r_ » 1 ( v +lf c i — ; ) v a ( 1 " a ) + a b ^ Where a -- 1 - V f 2 "2Dg"~ b = W Clearly, since f 3 and f 4 must both be positive and since vm must be less than v a , vm a \ £ va ~ l / v a 2 ( 1 " a ) + a b \ 4 ( 5 ) The slope of the curve vm = f ( v a ) a t ^ p o i n t i s dv. ( ) m - 1 (1 - (1 - a) V 4(6) dv a a ( y (1 - a)+ ab/v| ) The concavity of this curve may be assessed by taking d vm - - (1 - a ) b 4(7) d v ^ (v a 2 ( 1 - a)+ a b ) 3 / 2 Note that where v m =0, equations 4(5) and 4(6) reveal that and 54 At very large a i r v e l o c i t i e s , - 1 -M 1 V a #>co f Where 4 = 0 , a = 1, the curve i s a s t r a i g h t l i n e with a slope f equal to 1. Where 4 i s large such that a approaches zero, or f ah where 3 approaches zero, — ~ ^ n equation 4(6) approaches zero f a s t e r than a, l i m a = 0 y i - a = 1 a 2 and, therefore, ^ = 1/2 3=^0 :;, The above r e s u l t s may be summarized as shown i n Figure 8, where l i n e s A and B correspond to the two l i m i t i n g cases f o r which d vm i s equal to 1.0 and 1/2 r e s p e c t i v e l y , dv a Also shown i n Figure 8 are the r e s u l t s of Richardson and McLeman and Hitchcock and Jones. These published r e s u l t s seem to be i n accord with the t h e o r e t i c a l analysis of the s l i p r a t i o presented i n t h i s t h e s i s . Note that the slopes of a l l the curves are within the l i m i t s imposed by equation 4(3), as defined by l i n e s A and B i n Figure 8. The above curves apply to h o r i z o n t a l flow; i n the case of v e r t i c a l flow f o r the same p a r t i c u l a t e m a t e r i a l , the curves would no doubt be displaced to 55 the r i g h t of those f o r h o r i z o n t a l flow and, i n the case of curved ducts, an increase i n v e l o c i t y would increase drag, reduce the s l i p r a t i o , and displace the curve downward, probably to values of minimum slope l e s s than 26j£°. o io 20 : 30 40 50 60 TO 83 9 0 Too fio fao J30 AIR V E L O C I T Y \£ = f r / s e c Fig.8 V a r i a t i o n of Material V e l o c i t y with A i r V e l o c i t y i n Simple -Horizontal Flow i n a 1" Duct Values of f ^ and f ^ may be obtained f o r the experimental data shown i n Figure 8. These values are derived from equation 4(4) and are shown i n Table I I I . The r e s u l t s i n Table I I I are anything 56 "but conclusive, p r i m a r i l y because several assumptions regarding drag c o e f f i c i e n t s and a i r d e n s i t i e s had to be made i n the case of the glass p a r t i c l e s and the peas, but p a r t l y because the accuracy of the free f a l l i n g v e l o c i t i e s f o r the other p a r t i c l e s i s unknown. However, these r e s u l t s w i l l be h e l p f u l l a t e r i n the evaluation of f ^ and f ^ i n connection with the r e s u l t s presented i n t h i s t h e s i s . TABLE I I I FRICTION FACTORS CALCULATED FROM LITERATURE DATA P a r t i c l e s f ^ f ^ Aluminum 0 2 . 0 0 Coal 0 2.30 Lead 0 1.26 Peas 1 .50 x 1 0 ~ 5 0 Glass 1 .20 x 1 0 " 5 0.221 DETERMINATION OF VELOCITY FOR MINIMUM PRESSURE DROP IN A DUCT Equation 3(50) s i m p l i f i e s considerably f o r a s t r a i g h t h o r i -zontal or s t r a i g h t v e r t i c a l duct, such as those considered by Zenz and Rose et a l . For a h o r i z o n t a l duct of length L, where co m p r e s s i b i l i t y and p a r t i c l e a c c e l e r a t i o n e f f e c t s are ignored APH = (f^^2* WavVa1> L + W » KL » V.V \ 4(8) AD 5 B i C DA pA d ( v a S s — 5 5 5 ) 57 Similarly for a v e r t i c a l duct, ^ v = ( W a 2 + W H V + W m ( & L + V a V - ] 4(9) AD ° D A p V ( v a S H 2 S D >" If equation 4(8) i s differentiated with respect to velocity, a l l S S other quantities being held constant, and noting that av = H, d ( ^ ) = ( 2 f ^ a v a + WH>- L + f2 Sp Wm C- f3 L • + F4SHL ) dv A D 2Dg C DA pA D ( V a 2 g H "~^S^~ ) For the optimum velocity, with respect to pressure drop, the following cubic equation, which has one real and positive root and two negative or complex, may be solved. f f i e a > a 3 + <WH + f 2 W A j T a 2 - } W » H = 0 4(10) I Dg ] ^ D g " 2C DA pA DDg j ^DVDSH) Similarly, for v e r t i c a l ducts, equation 4(9) becomes: | £ l i a | V a 3 + f W v + W W v ^ v a 2 - f W a ) = 0 4(11) | D g > ^2A^ D T 2CDI pA DDg j ^ D A p A D S v ) In the case of v e r t i c a l ducts, equation 3(27) indicates that f^ i n equation 4(10) would be replaced by B±n'=K= 1.0. But f^, the coefficient of sli d i n g f r i c t i o n , should never approach this value. Therefore, i t would be expected that the optimum velocity for v e r t i c a l flow would be higher than that for horizontal flow, just as shown by the above-mentioned writers. If f u l l informa-tion on the tests conducted by these other writers were known, i t might be used, in conjunction with the evaluations of f , and 58 f 4- given i n the previous s e c t i o n of t h i s chapter, to deter-f mine values of 2 f o r v e r t i c a l and h o r i z o n t a l pneumatic con-veying. However, such complete information i s not av a i l a b l e and, i n any event, considerable future research i n t h i s area f f i s j u s t i f i e d with the s p e c i f i c purpose of r e l a t i n g 3 , 4 and f 2 to p a r t i c l e c h a r a c t e r i s t i c s . I t should be noted that the so - c a l l e d "optimum" v e l o c i t y as i t i s r e f e r r e d to i n t h i s t h e s i s i s optimum with respect to pressure drop i n a duct and may not be s a t i s f a c t o r y i n s o f a r as system r e l i a b i l i t y i s concerned. Zenz has shown that, p a r t i c u -l a r l y i n the case of h o r i z o n t a l ducts, the region of optimum v e l o c i t y i s extremely unstable. Prom a p r a c t i c a l standpoint, i t would be well to use i n i t i a l a i r v e l o c i t i e s s u b s t a n t i a l l y above the optimum v e l o c i t y c a l c u l a t e d by the foregoing procedure . SUMMARY f f Although f u r t h e r considerations of 3? 4- and optimum v e l o c i t i e s f o r minimum pressure drop i n ducts of any shape and geometric c o n f i g u r a t i o n might e a s i l y be made, no purpose would be served by doing so at t h i s time. Therefore, l e t i t s u f f i c e to say that the work of Zenz, Rose et a l , Richardson and McLeman and,Hitchcock and Jones, although not c o n c l u s i v e l y v e r i f y i n g the t h e o r e t i c a l considerations presented i n Chapter 3 , at l e a s t lend s u b s t a n t i a l support to those considerations and j u s t i f y and encourage the experimental research h e r e i n a f t e r presented. CHAPTER 5 EXPERIMENTAL APPARATUS Early i n the course of this research, i t was decided that the most important variable i n the design of pneumatic conveyors i s the pressure drop from one end to the other. The i n i t i a l point i s usually taken to be the point at which particulate material i s injected into the a i r stream. Equation 3(50) i s essentially an expression for determining the pressure drop attending two-phase compressible flow i n a closed duct. However, before equation 3(50) can be used with any confidence, f f f the values of the three f r i c t i o n factors, 2, 3 and 4, must be determined for given duct and material characteristics. Any relationship between these f r i c t i o n factors and other system variables, such as velocity and air-to-material ratio, must also be determined. Fortunately, from an experimental standpoint, a l l of the variables i n equation 3(50) can be measured i n the laboratory. The apparatus used i n this research was designed with a view to varying and measuring as many of these variables as possible. The design and operation of the various components which made up the apparatus w i l l now be discussed i n d e t a i l . A l i s t of component specifications i s given i n Table A of Appendix I. 60 THE DUCT Economics and time dictated the use of one duct config-uration and diameter. However, the duct used provided con-siderable variation i n conveying conditions throughout i t s length. The duct size used was 2" standard pipe, having an inside diameter of 2.067"° The basic duct layout i s shown i n Figure 9, together with the manometer schematic and pressure and temperature tap details. The distances between manometer taps and other information about the duct are given i n Table B of Appendix I. The in s t a l l a t i o n of duct from point 9 to point 15 was accomplished by previous students; i t was considered expedient to make use of this previous work. The numbered points along the duct, as shown i n Figure 9? are pressure taps. Frequent use i s made hereinafter of these numbers as subscripts to represent pressures, temperatures, and velocities at their respective points. A l l of the elbows i n the duct had a three-foot radius to their centerline and were flange-connected to the straight sections of the duct. The pressure taps at the discharge end of each elbow, such as "10 and 11, are two feet apart. Reference w i l l occasionally be made to the elbows by stating the number of the pressure tap at the elbow i n l e t , such as elbow 15. Elbows 6, 9, 12 and 15 l i e i n a ver t i c a l plane, whereas elbow 18 l i e s i n a horizontal plane. 87^0" • -IS • -14 II \0, 2" S T D . P \ P E 7 ^ 9 0 E L B O W NO'.S 1TO 21 A R E P R E S S U R E A N D T E M P E R A T U R E T A P S 2.1 A a. 3) V C Y C L O N E 8+ F E E D E R ^ , 0 R | F | GLASS ELBOW ^  ^ G L A S S S E C T I O N 6 5 J S Y S T E M S C H E M A T I C war* 61 BLOWER 5 fc 7 8 9 toil I2B I4 ISI6I7\8I9 232I , , , 9 \2. © 4 tmttmmwmm WW M A N O M E T E R S P L A S T I C HOSE.-* F I L T E R WHITE TEFLOM SHIELD T E M P E R A T U R E t P R E S S U R E T A P S Fig.9 System Schematic and Instrumentation. 62 The glass section of the duct from point 4- to point 7 was installed so that the acceleration characteristics of the particulate material could be visually observed. It also served as a safety measure: at low f l u i d velocities, any material saltation or slugging tendency could be readily observed at points 6 and 7» The p o s s i b i l i t y of using photo-graphic means to measure the velocity of the particulate material at various points between points 4- and 7 was also considered. The use of this technique, however, was not required. The pressure taps at the various numbered points along the duct were equipped with f i l t e r s , as shown in Figure 9° Unfortunately, a l l of the existing connections at these taps, primarily at the f i l t e r s , leaked and were necessarily modified to suit the existing bastard-size hose to standard f i t t i n g s . Each pressure tap was subsequently found to be leak-proof. Static pressures were measured at the tap points. It was sus-pected that several of the tap f i t t i n g s projected s l i g h t l y into the duct int e r i o r . However, the effects of these variations i n tap characteristics were not dete ctable with the manometers used, and these effects doubtless had l i t t l e effect on the overall results. But, even though these faulty taps were not modified, i t i s recognized that static flow pressures can only be measured accurately i f the taps deform the flow boundary as l i t t l e as possible. 63 Bimetal d i a l thermometers were used at points 0 and 3 , with the necessary 2 inches of stem immersed i n the f l u i d stream. The taps f o r these thermometers were comprised of sealed standard pipe f i t t i n g s . The presence of p a r t i c u l a t e material i n the f l u i d and, hence, the p o s s i b i l i t y of plugging, together with the need f o r remote temperature measurement at point 21, necessitated the use of thermocouples with s p e c i a l p r o t e c t i v e taps at points 3 and 21. The tap f i t t i n g s were made from t e f l o n . I t was assumed that the s l i g h t p r o j e c t i o n of these f i t t i n g s into the f l u i d stream provided good f l u i d mixing i n the protected v i c i n i t y of the thermocouple junction. THE BLOWER The pressurized f l u i d i n two-phase flow i s normally pro-vided by a pump. In the case of pneumatic conveying, the a i r supply i s u s u a l l y a compressor, blower, or fan. Where the second phase i s a l i g h t m aterial, the duct i s la r g e , and the a i r pressure i s below one p s i , a fan i s often used; phase mixing occurs at the fan i n l e t . In the case of heavier mater-i a l s , small ducts, and a i r pressures over one p s i , i t i s impera-t i v e to have an a i r supply which does not vary excessively with pressure; consequently, a p o s i t i v e displacement compressor or blower i s normally used. The Roots blower i s frequently employ-ed i n t h i s type of a p p l i c a t i o n . Phase mixing i s normally accomplished by an i n j e c t o r at some point downstream from the blower. 64-The blower available for this research was a small Delco Remy blower of the Roots design. Its origin was somewhat untraceable, although i t appeared to have at some time served as a supercharger on a diesel engine. However, the positive-displacement features of this blower at a i r pressures over 8 psi seemed well suited to the needs of this research. The blower had been f i t t e d previously with a 10 hp 1760 rpm motor. Unfortunately the motor base had been improperly fabricated and installed, thus providing very poor alignment between the blower and motor. O i l in the area surrounding the blower suggested that previous operation with this misalign-ment had already damaged the seals on the blower drive shaft. Drive belt damage also contributed to this conclusion. Conse-quently, the blower drive base was modified and the blower and motor properly aligned. Other modifications to the previous i n s t a l l a t i o n consisted of f i t t i n g the blower and motor with a variable speed drive; f i t t i n g the blower with a silencer, i n deference to people work-ing i n the immediate area; and modification of the blower discharge port to ensure no a i r would be lost at that point. The finished i n s t a l l a t i o n , showing the blower, blower motor, and silencer i s shown i n Figure 10. T r i a l runs showed that the maximum blower speed which could be attained without the blower leaking o i l excessively was about Fig.10 Blower, S i l e n c e r and Motor. 66 1800 rpm. A major overhaul of the blower would have been required to replace the seals and, thereby, raise this upper lim i t of blower rpm. Time did not permit such an overhaul and, fortunately, the requirements of this research were ade-quately f i l l e d at blower speeds below 1800 rpm. Ammeter readings established the blower discharge pressure limits above which the motor would kick out. The lack of f u l l information and specifications on the blower required that some means be found to measure the blower discharge rate. An o r i f i c e plate had previously been installed at a point a short distance downstream from the blower. Preliminary test runs showed, however, that the presence of the or i f i c e plate i n the duct created a pressure d i f f e r e n t i a l which was incompatible with the aims of this research. In other words, the maximum pressure attainable downstream from the o r i f i c e , when the pressure upstream was below the previously mentioned li m i t s , was below that required for the planned experiments. Moreover, the variable discharge characteristics of the blower at low pressures were further aggravated by the presence of the or i f i c e i n the duct. It should be noted, too, that since the measurements involved i n the contemplated test-runs were to be taken by one man, the elimination of direct reliance on o r i f i c e measurements seemed desirable. The above disadvantages of an o r i f i c e plate were compounded by the fact that a lack of blower performance characteristics 67 made i t almost impossible to preselect the blower speed for a required delivery rate. Both of these d i f f i c u l t i e s were overcome by relating the delivery characteristics of the blower to discharge pressure and rotor speed. This was done during preliminary test-runs . The discharge rates were measured at various blower speeds, using two orif i c e s of d i f f -erent diameters, to cross-check the performance of each o r i -f i c e . The downstream pressure tap was located such that the pressure would be measured i n the vena-contracta region of the or i f i c e discharge. 10 In Flow Measurement with Orifice Meters , i t has been shown that the flow rate through an o r i f i c e may be determined primarily i n terms of the pressure drop across the o r i f i c e . S p e c i f i c a l l y V a - 359.1 ( D o ) 2 c x K Y ./^VV 1b/hr° 5(1) Where Dx. • o r i f i c e diameter, i n . o ex. - expansion correction for o r i f i c e K = flow coefficient (>yj .» f l u i d density at o r i f i c e , l b / f t ^ w = pressure drop across o r i f i c e , in-B^O I = expansion factor = 1 - (0.41 -8- 0<,35 K4).AP PK For this research, A P *=* constant P K - 0.-50 for 1.00" o r i f i c e = 0.709 for 1.418" or i f i c e 68 ex. = 1.0 Y = 0.98 Therefore, the flow rate through the one-inch o r i f i c e was taken to he V = 0.211 F 1 n g 5(2) T1 and the flow rate for the 1.418-inch o r i f i c e was taken to he W a » 0 .603 5 ( ? )  T1 Where i s i n "h^ n g * P 1 " P 2 ~ "^g * P r e s s u r © drop across o r i f i c e T^ = degrees Rankine The experimental data taken during the blower ca l ibrat ion test -runs are shown i n Table C of Appendix I . It i s known that a posit ive displacement blower has a volumetric eff ic iency which i s related to the blower pressure and rotor speed. Consequently i t i s reasonable to assume that the discharge rate w i l l be the rotor speed times the rotor displacement, less the loss due to what i s often cal led blower s l i p . The blower s l i p i s usually taken to be a function of discharge pressure. Hence, the f o l l -owing function was chosen to represent the blower discharge rate . V a " ?a { ^ b + b + - c P 1 + ^ l 2 + " " ] 69 A simple but tedious curve f i t t i n g process r e s u l t e d i n the following expression, ignoring pressure terms of higher order than 2. IT F a t (N, + 11.4 P . 2 - 219Px, - 292)lb/sec. 5(5) W a = 8 6 ^ ( b 1 1 ) The use of equation 5(5)» wherein P^ i s i n p s i g , P & ^ i s i n p s i a , N^ i s i n rpm, and T ^ i s i n degrees Rankine, provided c a l c u l a t e d values of W& f o r a l l the cases i n Table C of Appendix I. The c a l c u l a t e d values are p l o t t e d against the measured values i n Figure 11. Figure 11 indicates that a second order polynomial provides a very good measure of blower s l i p . The determination of an expression r e l a t i n g blower discharge rate to r o t o r speed and discharge pressure permitted the removal of the o r i f i c e p l a t e from the duct. Equation 5(5) was used to provide a set of general curves r e l a t i n g blower discharge rate to r o t o r speed and discharge pressure. These curves are shown i n Figure 12. Note that the operation of the blower was r e s t r i c t e d to the left-hand side of the cross-hatched area i n the f i g u r e due to the motor l i m i t a t i o n s . These curves were extremely h e l p f u l i n evaluating the performance of the o v e r a l l experimental arrangement as well as i n s e l e c t i n g the range of operation f o r test-runs. In a d d i t i o n , equation 5(5) was easier to use i n c a l c u l a t i n g w*a than would have been the general o r i f i c e equations. A simple modification of equation 5(5) provided a new 70 0.30 0.25 jj 0.20 <h O.IO < Ul o-os OOS O.IO OI5 0.20 0-25 0-30 W a = + II.4R* - 219 F! - 28Z) lb/sec F i g . 1 1 Accuracy of Function Used to Compute W over Pertinent Range. 71 equation for the air velocity at point 1, where v a^ i s i n OT ft/sec. and i s i n R. 0.45 Q40 > a S 0 . 3 5 J ui o ^  0 . 3 0 CL Id 5 o j 0-25 CD I. O.I5 OWO 0.05 \ = Fgt ( N + I .4P*- 2 O R - S 92) Jk 'sec N = F F?=f !PM >sig Ta« 5: .7 psia ;S*R N •IO HP TING OF Tl CURV RANS IIS ct/» L - OP E TO i VE :RA-LEPT ////// '////// W/A \ /J77/f > ///// 3 4 5 P, * p s i g 6 8 IO Figo12 Blower Delivery W vs. Discharge Pressure P,j for various Blower Speeds. v a 1 = T 1 P a t (N.+ 11.4P1 2- 219P* - 292 )ft/sec. 5(6) , D 1 } 5 - 0 6 ^ (Again, the use of this equation and the family of curves i t provided, as shown i n Figure 13, provided a very convenient 72 means f o r determining the desired blower speed f o r given t e s t requirements. Another important r e v e l a t i o n contained i n Figures 12 and 13 i s that the Delco Remy blower has considerably l e s s than constant displacement at low pressures. The discharge rate at Fig.13. A i r V e l o c i t y at Feeder I n l e t vs. Blower Discharge Pressure at Various Blower Speeds. pressures over 6 p s i i s f a i r l y constant. This i s not too sur-p r i s i n g , inasmuch as the blower was designed f o r operation at pressures over 6 p s i . In the case of t h i s research, and the experimental procedure 73 which w i l l he revealed presently, the v a r i a b i l i t y of the d i s -charge rate at low pressures proved to be an advantage. As the material i n j e c t i o n rate increased, the pressure rose, thus reducing the blower discharge r a t e . Consequently, at one blower speed, a number of i n j e c t i o n rates provided a wider range of f l u i d v e l o c i t i e s than would otherwise have been p o s s i b l e . THE INJECTOR A constant displacement blower or compressor requires that the p a r t i c u l a t e phase i n pneumatic conveying be i n j e c t e d into the a i r a f t e r i t has been compressed. The most common type of i n j e c t o r i s the r o t a r y a i r - l o c k or pocket-rotor feeder. I t normally consists of a housing i n which revolves a r o t o r with pockets. Two s i g n i f i c a n t disadvantages i n t h i s type of i n j e c t -or are i t s removal of compressed a i r from the duct and the ex-cessive wear which occurs between the r o t o r and the, housing. The i n i t i a l a i r l o s s i s a t t r i b u t a b l e to the revolving r o t o r pockets; f u r t h e r a i r losses occur due to wear between rotor and. housing. These disadvantages prompted a search f o r another means of i n j e c t i n g non-free-flowing s o l i d s into a pressurized a i r stream. This search l e d to the development of a new type of i n j e c t o r . The i n j e c t o r b a s i c a l l y consists of four r e s i l i e n t r o l l s which are engaged along l i n e s of contact and, hence, turn together. Pig.14- I n j e c t o r I n s t a l l e d . 75 The four r o l l s form an enclosure between them, through which the a i r stream passes. P a r t i c u l a t e material i s fed into the a i r stream through one of the inward-turning r o l l e r i n t e r f a c e s , much as clothes are f e d through washing machine wringer r o l l s . Figure 14 shows the i n j e c t o r i n s t a l l e d i n the duct, at the bottom of the c o l l e c t o r chute. Provided the seals are properly sealed, and provided the r o l l e r material i s adequately r e s i l i e n t , the a i r losses i n t h i s type of i n j e c t o r should be small compared to those inherent i n a pocket-rotor feeder. Figure 14 shows the i n j e c t o r drive arrangement. A Carter Gear variable-speed drive was driven by a 3/4 hp motor. The Carter Gear, i n turn, drove a small worm gear through a v a r i a b l e v - b e l t d r i v e . F i n a l l y , the worm gear drove one of the top r o l l s i n the i n j e c t o r . The r o l l e r speed range was from 0 to 300 rpm. DRIVING THE ROLLERS Although the i n d i v i d u a l r o l l s i n a f o u r - r o l l feeder turn, normally, without slippage occurring at t h e i r various l i n e s of contact, the angular v e l o c i t y of each r o l l depends upon i t s s p e c i f i c diameter. Any v a r i a t i o n i n r o l l e r diameter between the r o l l s requires that the r o l l s turn at d i f f e r e n t speeds which vary accordingly, i f no s l i p p i n g i s to occur. Consequent-l y , i t i s not p r a c t i c a l to t r y to drive more than one r o l l at a constant speed, such as would be the case i f a single chain 76 drive with a sprocket on more than one. r o l l were used. I t i s conceivable that some kind of compensating device might be used to take v a r i a t i o n i n r o l l e r diameter into account, but t h i s would e n t a i l a formidable increase i n the o v e r a l l cost of the apparatus. I t follows, then, that the apparatus must be driven through one r o l l . Unfortunately, the f r i c t i o n between the r o l l s and the end-seals i s considerable, even when a l u b r i c a n t i s used, and the torque, therefore, i s considerable at each r o l l . Moreover, i f one of the top i n j e c t i n g r o l l s i s the one that i s driven, i t s mate requires a d d i t i o n a l torque to overcome the resistance of the pressure within the annulus against the material being i n j e c t e d . The torque requirements at a l l r o l l s other than the one which i s driven d i r e c t l y must be s a t i s f i e d by f r i c t i o n along the l i n e s of contact between the r o l l s . The pressure along the l i n e of contact, or area of contact, since the r o l l s are s l i g h t l y compressed, must be high to provide the necessary f r i c t i o n , thus c o n t r i b u t i n g a major source of r o l l e r wear. Apart from the problem of r o l l e r wear, material which adheres to the r o l l s reduces the contact f r i c t i o n , more or l e s s because the adhering p a r t i c l e s act as r o l l e r s or b a l l s , to something only s l i g h t l y greater than r o l l i n g f r i c t i o n . The i n j e c t i o n of Maw seed, which i s s p h e r i c a l and 0 . 0 3 2 " d i a -meter (approximately) r e s u l t e d i n ; a l l but the d i r e c t l y driven r o l l ceasing to turn. Much the same r e s u l t was obtained with 77 sand-blast, which i s spherical but has a wide distribution of diameters, the mean being about 0.0207. In the case of sand-blast, the torque requirements at each r o l l were considerably-greater than under normal conditions because the smaller part-i c l e s became lodged between the seals and the ends of the r o l l e r s , eventually causing the feeder motor to kick out a c i r c u i t breaker with only the dire c t l y driven r o l l turning. Each r o l l e r was equipped with two metal washers, one on either end of the rubber body. The washers at the four r o l l s were i n r o l l i n g engagement at each end. The transition of the injector enclosure from and back into the duct was accomplished at the i n l e t side with a fluted insert which f i t t e d between the r o l l e r washers, thus eliminating turbulence at that point, and at the discharge side by a diamond shaped transition which en-closed the entire injector opening. Brass sealing rings, fixed to the injector base, were held by adjustment screws against the r o l l e r washers. Sliding contact existed between the r o l l e r washers and the brass sealing rings. Machining errors produced an imperfect f i t between the sealing rings and the duct t r a n -sitions ; i n addition one of the r o l l s was perceptibly shorter than the other three. These d i f f i c u l t i e s were overcome to some extent by many hours of operation, during which time, the brass sealing rings became seated. However, these poor f i t s and the inadequacy of the seal ring adjustment screws, inasmuch as para- ' l l e l adjustment was well-nigh impossible to achieve, resulted in 7 8 a i r losses at the i n j e c t o r which were comparable to those experienced i n a pocket-rotor feeder. AIR LOSSES Before experimental t e s t s could be c a r r i e d out, i t was necessary to devise a method f o r determining the a i r lo s s e s . I t was found that when the brass rings were held against the r o l l e r washers with the maximum adjustment screw f o r c e , the r o l l e r s were locked but no perceptible amount of a i r was l o s t o Moreover, i t was found that the pressure losses i n the duct downstream from the i n j e c t o r conformed with those expected from the Darcy-Weisbach equation. I t was decided, therefore, to use the duct as an o r i f i c e to measure the flow rate i n the duct downstream from the i n j e c t o r . The flow rate ahead of the i n j e c t o r was determined, as previously shown, by measuring the blower speed and discharge pressure. This approach enabled the determination of the i n j e c t o r e f f i c i e n c y , ^ , f o r the case where no material was i n j e c t e d . Fortunately, i t was found that the a i r l o s s due to "blow-back" i n the i n j e c t i o n process was imperceptible and could be considered negligible compared to the losses at the se a l s . The r e s i l i e n c y of the i n j e c t o r r o l l s was such that the r o l l s maintained a t i g h t seal around the p a r t i c u l a t e material as i t was i n j e c t e d . The a i r losses due to i n j e c t i o n were evaluated by i n j e c t i n g 0.16 inch p a r t i c l e s ; a t a very low r o l l e r speed, 79 then observing the pressure differentials between points 1 and 3, and 4 and 5» in. inches of water. These pressure drops were not measurably affected by the presence of particulate material between the injecting r o l l s . As a rough check, no blow-back could be visually observed or f e l t above the r o l l s when particu-late matter was between the r o l l s , whereas the a i r losses at the seals, were quite apparent to the senses. Convenience demanded a means whereby the air losses could be readily measured and checked during the course of experimental test-runs. The simplest method which seemed available was to treat the duct sections between pressure taps 1 and 3 and 4 and 5 as o r i f i c e plates and measure the pressure drops across these lengths of duct. Then the flow ratio of these pressure d i f f e r -entials could be correlated with the injector efficiency. From the basic o r i f i c e theory and equation 5 ( 1 ) » w a < * l/e a\ Therefore, Since T 1 < a . T ^ W. W a4 a1. & \ 4 P 4 A P 4 - 5  P 1 A P 1 - 3 5(7) Rather than rely on equation 5(7)? i t was decided to conduct a series of tests with various seal pressures and blower speeds 80 and correlate the parameters Wa4 versus P 4 a P 4 - 5 Wa1 The results of these tests are shown i n Table D of Appendix I. 1.00 o.9o 6 Z. O.0O IL UJ 0 7 0 or UJ 0 0 . 6 0 ul ul 0 5 0 ti-ll 0.40 •dm s& 0 3 0 ll O.2o O.IO St A i A L i N O i > U) IGH iR»C r ATEC p C T = 0 8o ] S" D 3 ' S E A L S DR V Al ID L oos\ OO I84.< I30C > BL il 5 RF ll M D 1SOC \5ec 1 <l i\ il i< ® 169< ) II <l M o r e : F E E D E R LOS SES « 0 O-l 0-2 0-3 0-4 O S 0-6 0 .7 0-8 °-9 I . O I.I 1,2. 1-3 1.4 L S A R - 3 P , Fig 0 1 5 Feeder Efficiency vs. a Measurable Pressure Parameter. These results are plotted i n Figure 15 . Values of over 0„80 were obtained with the seals lubricated. A number of different lubricants were t r i e d , but 3 ^ ^ . . 2>o o i l was the only one which did not become sticky and binding. Unfortunately, the 81 thirty-weight o i l required continual replenishment, undoubt Equation 5(8), although similar, i s not equivalent to equation 5(7). The dissimilarity between-equations 5.(8) and 5(7) i s no doubt due to the assumptions which were made in simulating two or i f i c e s with sections of duct, one of which contained a short radius elbow. In any case, for the purposes of this research, Figure 15 and equation 5(8) were deemed satisfactory for deter-mining injector air losses. At the same time, i t was decided to avoid the use of lubricants for purposes of convenience. Injection Characteristics A method having been established for evaluating injector efficiency, attention was turned to the injection process i t s e l f . Preliminary tests on an old washing machine wringer indicated that the injection rate was a function of material head over the r o l l s , particle size, sphericity, bulk density, solid density, and r o l l e r radius, length, speed and resilience. In selecting the r o l l e r surface, f l e x i b i l i t y in operation was the main cri t e r i o n . The f i r s t attempt entailed the f a b r i -cation of pneumatic tire s from 3-inch diameter hose and pieces edly because i t was vaporized by escaping a i r . A log-log plot of these results provided the curve 5(8) 82 of neoprene. This attempt was plagued with problems t the contact cement used in fabricating the t i r e s was inadequate, several punctures occurred during t r i a l runs, and a t i r e pressure sufficient to withstand the pressure inside the annulus caused distortion i n the t i r e contour. Inevitably, as i t were, sol i d neoprene tires were vulcanized in place and machined to the correct diameter. These t i r e s , although restricted to one t i r e pressure, provided excellent service throughout the experimental work. Figure 16 shows a cross-section through the injector. The clearance between the chute above the injector and the top r o l l e r s was about 1/32". This restricted operation of the injector to particles of diameter greater than 1/32". If a given particulate material rested on the top r o l l s , under a head of h ( in Figure 1 6 ) , and i f the r o l l s were turned inward-l y , the particulate material would compress the r o l l s such that an apparent opening of width t would appear between the r o l l s . The volumetric feed rate per revolution for a fixed value of h would then be F* -. 2TTr_L t r r Preliminary tests on a washing machine wringer indicated that t was related d i r e c t l y to r r , ^ p / ^ t and to a dimensionless parameter containing and h. One such dimensionless para-meter was found to be 83 Hence, 5(9a) Fig.16 C r o s s - S e c t i o n Through I n j e c t o r The constants a and b were determined by t a k i n g average v o l u m e t r i c feed r a t e measurements f o r v a r y i n g ho F OT = 1 h—* mx r hmx F fdh 8 4 a V A 2 ? L -fash I* + 1- r i r r (.xb° mx_; Now, f o r the i n j e c t o r used i n t h i s research, many of the varia b l e s i n the above equations were held f i x e d such that 5(9b) a •-' 3 a r -T Ii, r r r ( r T?) r r xb + 1 ) = constant ' 2 a r _ L / „ sb r r Hence, , K ) ll?] 5 ( 1 0 ) and Equation 5(9b) becomes 2 F ^ * av mx b + 1 ( 5(11) The constants b + 1 and a\b+ 1 ) were determined from the data shown i n Table E of Appendix I» These data are p l o t t e d f o r c e l l u l o s e beads and vetch seed i n Figure 1?« From Figure 17 i t was found that, f o r c e l l u l o s e beads, b + 1 - 1 «145 and a = 28o30» The volumetric feed rate then becomes = V ' . 28.3 N ± C j , <fb V ).-1*5 iaV-lnata 1 and the mass feed rate i s simply ( ) .145 WM = 28.3 N ± e P ) N. lb/minute 85 5(12h) X > l 6 2 8 7 6 5 4 3 |6* Z 7 5 4 3 2 10" 8 7 6 4 3 7= PU STIC rCH BE/ SE.« \0'. ID 3 © or* e- i _ 71 3 1 »- / -T £ 3 F i g d 7 Log Plot of Injection Parameters The apparently constant value f or b (for the materials tested) regardless of material characteristics simplifies the 86 use of equations 5(12a) and 5(12b). From equation 5(11)? T?' Q 2 h U 2  a = av v b nmx (1.14-5) C ^ i 7 5(15a) ^ p N i 2 Wmx 3' and, by d e f i n i t i o n , , Volume Over Injector @ h ^ Time to Empty Hopper x r o l l e r speed = Vhmx 5(13b) e i Then s u b s t i t u t i n g 5(13h) i n 5(13a), 5(13a) i n 5(10), and 5(10) i n 5(12a) and 5(12b), Q m = vhmx in^/seco 5(14) V m = 1 a 1 Z <"5 Vhnx&o lb/sec. 5,(15) t e where ^hmx i s the hopper volume i n inches cubed and ^ e - the time to empty the hopper i n seconds. Equations 5(14) and 5(15) make i t possible to measure the feed rate from steady state conditions by merely i n t e r r u p t i n g the c i r c u l a t i o n of material and timing the hopper emptying time. C l e a r l y , t h i s i s a much more convenient method of determining m^ than would be afforded by the use of equations 5(13)» This discussion of the i n j e c t o r would be incomplete with-out some comments on the propensity inherent i n the apparatus 87 to eject particles through the out-turning r o l l e r interfaces. A glance at Figure 14 w i l l i l l u s t r a t e this. It was hoped, originally, that the material would be cleared from the i n -jector annulus before i t would have an opportunity to be trapped between either of the two out-turning pairs of r o l l s . Although this i s the case for most materials tested, several notable exceptions exist. The tendency of certain electrostatically charged and o i l y materials such as Maw Seed to adhere to the neoprene r o l l e r s has already been mentioned. A large proportion of such materials injected into the enclosure i s automatically carried out of the enclosure. Lead shot (numbers 2 and 8), although not adhering to the r o l l s , created another problem associated with injection. An occasional pellet would lodge i t s e l f between the out-turning washers on the downstream end of the r o l l s . Once so lodged, the pellets would be rolled f l a t . Unfortunately, this f l a t t e n -ing process threw a severe strain on the injector transmission components, thus ruling out the use of lead shot i n this research for purposes other than merely qualitative observation i n the glass section of the duct. For the materials used in this research, in the study of the theory presented in Chapter 3, the discharge or ejection rate was found to be less than half a percent of the injection rate and was therefore ignored. However, future use of this apparatus might well warrant a slight modification to the hopper over the 88 Pig.18 Cyclone Receiver at the End of the Duct. 89 injecting r o l l e r s and,, perhaps, the i n s t a l l a t i o n of scrapers or guards within the annulus to prevent the ejection of materials through the r e s i l i e n t portion of the out-turning r o l l e r s and entrapment of particles between the out-turning metal end-washers. RECEIVER, SUPPORTING TOWER ANT) COLLECTOR CHUTE Two-phase flow i s usually used to transport a particulate material between two points. At the destination point, the phases must be separated before the particulate phase may be handled by other more conventional methods. Pneumatic conveyors normally employ a cyclone, or receiver, for this purpose. Figure 18 shows the cyclone used in the present research. This cyclone discharged the particulate phase into the collector chute di r e c t l y below i t and discharged the a i r to atmosphere outside the laboratory. The design details of this type of receiver are so common that a f u l l discussion thereof i n this thesis would be redundant. I t might be noted, however, that in a research i n s t a l l a t i o n where the particulate phase i s to be continually circulated, this type of receiver i s very damaging to such particulate material as vetch seed. In fact, the degen-eration of such materials, when continually circulated, v i r t u a l l y ruled out their use i n this research. The supporting tower had been erected during the course of Fig. 1 9 Supporting Tower and R e c e i v e r . 91 previous research. It consisted, essentially, of four dexion columns with cross-bracing at several levels, as shown i n Figure 1.9 • The receiver was supported at the top le v e l . A collector-chute with a deflector was designed so that the material flow from the receiver could be directed either back to the injector or into the weigh-barrel. A plexigl&s window V^LS installed at the bottom of the collector-chute so that the injection process could be visu a l l y observed and the material head over the injector r o l l s measured. Particulate material i n the weigh-barrel could be taken out of the system or returned to the injector via a poly-ethylene tube and a hinged door i n the collector chute, just above the plexiglass window. Of course, different materials were introduced into the system through this hinged door. The production of dust in the operation was held to a minimum by the use of polyethylene between the top of the weigh-barrel and the collector-chute discharge duct and by the close f i t s i n the various system components. An extra level of 3-inch channel was added to the bottom of the tower. This level served as a supporting platform for the injector and the injector drive components. This level of bracing also served as a guard, albeit a very scanty one, around the blower drive components. It was impossible to procure guards around the various drive components of the experimental apparatus without f u l l y detailed engineering drawings. 92 The scarcity of time and the impracticability of producing such detailed drawings for drive guards resulted i n the ris k being to l i f e and limb/endured, although only with .difficulty.. INSTEUMENTATION The manometers used for pressure measurement had been assembled during previous research. However, the connections and general manometer arrangement had been obliterated by people doing other research. It was necessary f i r s t to locate these manometers and then to assemble them i n a sensible array. The result of this work i s shown i n Figures 9 and 20. A l l the duct taps from points 5 to 21 are fed into a cen-t r a l manifold, from which the pressure at each tap may be measured on either a water or a mercury manometer, with the. reference pressure being either atmospheric or the pressure at taps 4, 9, 12 or 18. An additional water manometer was i n -stalled for continually measuring the pressure drop across the injector (between points 3 and 4). Another mercury manometer; was installed to measure the atmospheric pressures at points 0, 1, 2, 3 and 4, independently of the remaining taps. F i n a l l y a third mercury manometer was used to measure the pressure d i f f e r e n t i a l across the o r i f i c e plate (points 1 and 2) when the plate was used. Originally the manometers were equipped with pieces of 93 Pig.20 Manometer Arrangement. 94 surveyor's tape. Each pressure reading required two scale readings and the subtraction of one reading from the other: obviously an extremely i n e f f i c i e n t and time consuming process. Five s t r i p s of aluminum, a s c r i b e r , and endless patience produced scales which could be zeroed at the manometer miniscus and thereby provide each pressure reading with only one scale reading and no subtractions. The accuracy of readings taken from these manometer scales may be taken to be - 0 . 0 5 inches of water or mercury, whichever the case may be. The bimetal and thermocouple taps were discussed previously i n the se c t i o n on the Duct. The accuracy of the bimetal thermo-meter a f t e r being c a l i b r a t e d against a standard glass-mercury o thermometer over a range of from 60 to 180°F. was taken to be + ° - 1 .0 F. The thermocouple measurements were taken i n a d i r e c t reading potentiometer to an accuracy, again of - 1 .0 F. The thermocouple junctions were c a l i b r a t e d i n an a i r stream, ranging o o from 100 to 150 F., against the same glass thermometer used i n the c a l i b r a t i o n of the bimetal thermometers. Considering the influence of temperature on t h i s research and the d i f f i c u l t y normally encountered i n temperature measurement of f l u i d flow, the above-mentioned accuracies are considered s a t i s f a c t o r y . A strain-gage r i n g and weigh-barrel were av a i l a b l e from previous research. The b a r r e l was r e f i t t e d with metal supports and a s l i d i n g gate at i t s bottom. I t s f i n a l appearance i s shown i n Figure 21. An SR4 S t r a i n Indicator was used to measure the 95 Fig.21 Weigh-Barrel and Strain-Gage Ring. 96 signals from the strain-gage ring. The ring was calibrated from 0 to 30 lb. over the weight of the weigh-barrel. The calibration was essentially linear, i n accord with previous such readings, and can be stated simply as w s ^ f 5(16) where &R i s the difference between the Strain Indicator readings, i n microinches. The weigh-barrel and strain-gage ring were used to deter-mine the quantity of material i n the duct under steady state operating conditions. That i s to say, when injection ceased and the deflector i n the collector-chute directed the material in the duct into the weigh-barrel, the actual amount of material i n the duct during steady-state operation could be measured. The actual length of the duct and the injection rate during steady-state operations being known, i t was possible to deter-mine the average velocity of the particulate material i n the duct under steady-state conditions. Hence, V_ = weight of material i n duct at steady state, l b . L = actual duct length = 240 feet fit V M = injection rate i n lb/second and vmav = V m = 240 Vm 5(17) s s The accuracy of the strain-gage measurements was about = 1 97 micro-inch. Since the average reading of W s was greater than one pound, the e r r o r was l e s s than 1%, which i s quite accept-able . Various miscellaneous pieces of equipment were used during the course of the research and experimentation. For instance, a set of t o o l s was purchased f o r the p r o j e c t ; unfortunately most of the apparatus had been assembled before the too l s a r r i v e d . A micrometer was used to measure p a r t i c l e diameter. Bulk d e n s i t i e s were determined from the c a l i b r a t e d p ortion of the collector-hopper, together with the weigh-barrel. S o l i d p a r t i c l e d e n s i t i e s were determined by measuring the weight of a sample of p a r t i c l e s on a k n i f e balance and measuring the sample displacement i n a volumetric f l a s k p a r t i a l l y f i l l e d with water. Atmospheric conditions were measured on the laboratory standard barometer and thermometer. SUMMARY A summary of the apparatus s p e c i f i c a t i o n s i s given i n Appendix I. In t h i s chapter, the blower discharge character-i s t i c s have been analysed, the i n j e c t o r a i r losses have been expressed i n terms of a simple parameter, i n j e c t i o n character-i s t i c s have been established, and the design and operation features of the e n t i r e apparatus have been discussed. The treatment and c a l i b r a t i o n of the apparatus were performed with a view to p r a c t i c a l engineering and to convenience during the running of tests i n connection with the v e r i f i c a t i o n of the theory presented i n Chapter 3° CHAPTER 6 EXPERIMENTAL PROCEDURE The experimental procedure i n t h i s research was designed to use the apparatus discussed i n the preceding chapter, as f u l l y and advantageously as p o s s i b l e , i n the v e r i f i c a t i o n of the t h e o r e t i c a l considerations presented i n Chapter J . Although the s e l e c t i o n of p a r t i c u l a t e materials avai l a b l e had some bearing on the procedure, the f a c t o r which received primary a t t e n t i o n was the proposed method of data a n a l y s i s . This method i s d ealt with i n the f i r s t s e c t i o n of t h i s chapter. Subsequent sections deal with the p a r t i c u l a t e materials selected f o r the te s t s and the sequence and type of measurements taken during the t e s t s . PROPOSED METHOD OP DATA ANALYSIS A look at equation 3(50) indicates that the three unknown f r i c t i o n f a c t o r s , ^2^^ fee ^ r e a ' t e < i a s unknowns i n a l i n e a r equation. A set of three such equations would permit s o l u t i o n f o r these f r i c t i o n f a c t o r s . However, a set of 10 equations provides 120 d i s t i n c t combinations of three equations each. I f , i n f a c t , the f r i c t i o n f a c t o r s are constant over the experimental range from which the 10 equations are obtained and experimental error i s immeasurably small, the solutions of a l l 99 120 sets of equations should be identical. Where small ex-perimental errors exist, the solutions should s t i l l be nearly-identical. In the la t t e r case, an arithmetic mean and a standard deviation might be calculated i n the case of each f r i c t i o n factor. If any one or more of the f r i c t i o n factors i s not constant, then the solutions to the sets of simultaneous equations w i l l vary accordingly over the experimental range. In this case, i t f f w i l l be possible to investigate the variance of 2 when 3 and f 4 are held constant, i n accordance with the literature and theory correlation presented i n Chapter 4. In addition to the above evaluation of the experimental data with a view to direct v e r i f i c a t i o n of the theory and deter-mination of the f r i c t i o n factors, the experimental procedure should be designed to accomplish several other desired ends. For instance, the variation of s l i p ratio and polytropic gas constant ratios with air-to-material ratios should be studied. Furthermore, the l i n e a r i t y of the pressure gradient over the length of the duct should be evaluated. F i n a l l y , as much data as possible should be taken, bearing in mind the fact that one person i s to take the readings, and the limitations of the apparatus and the particulate materials to be run in the tests. 100 SELECTION OF PARTICULATE MATERIALS Preliminary tests were carried out on a. variety of materials i n an effort to find materials suitable to the purposes of this research.. The primary requirement i n a prospective material was a resistance to damage during con-tinual circulation i n the test apparatus. This requirement established the constancy of particle properties during the course of the tests. It should be noted that shape, size, and size distribution of the particles had l i t t l e bearing on the principal object of this research, namely the ve r i f i c a t i o n of the theory presented i n Chapter 3, provided these characterist-ics were constant throughout the tests. Among the materials considered for testing were those l i s t e d i n Table IV. The "preliminary results" were obtained during t r i a l tests. As shown i n Table IV, the cellulose pellets and barley kernels were the only materials which showed signs of behaving s a t i s f a c t o r i l y during a series of tests lasting several hours. However, these two particulate materials were deemed sufficient for the purposes of this research. It should be noted that although a minor amount of particle damage occurred i n the case of the barley, the dust and damaged kernels were ejected continuously from the system at the injector. 101 TABLE IV PARTICULATE MATERIALS TESTED N a m e Master S h a * e Solid Bulk Density Densit: Maw Seed ,032 Vetch Seed .124 Australian Winter Peas .208 Cellulose Pellets .162 Rice Barley .121 Lead Shot #2 Lead Shot #8 Saw Dust •=• Spherical .0358 .0254 Spherical .0456 .0270 Spherical .0456 .0280 Cylindrical .0326 .01730 El l i p s o i d a l E l l i p s o i d a l .0486 .0305 Spherical .045 Spherical .405 Angular .0250 .0070 Preliminary Results Broken, shells removed Broken, shells removed Broken, shells removed Excellent performance Broken kernels Excellent performance Jambed injector Jambed injector Severe ejection SELECTION OF VARIABLES FOR MEASUREMENT Before a decision could be made as to which variables were to be measured during the projected test runs, i t was necessary to rewrite equation 3(50) i n terms of the apparatus subscripts. That i s to say, for example, the i n i t i a l point i n equation 3(50) changes from #1 to #3« In addition, the i n i t i a l velocity must 102 be corrected for the a i r losses at the inje.ctoro Hence, a1 becomes va3c, Ca1 becomes (?a3» and so forth. Hence, ( 2 ) AP - (f1?a5va5c L + f4-WmSavva5cL ) (1 +AP) ( S5g 2DADg ) 2nFg + f2 Sp Wm lajc (1 +£Pj, + V a 5 c S a v L }(1 + AP) C DVD | S nP 3 2Dg < <\ 2W; \ T ( f 5 C 0 S g y s i n c 1 a ) (Xm+1 -xm) = AP ( xm^ 2° ^m) Um AP (2x_Q_ + © 2 ) 2 6(1) ( m - *)np mm j j Equation 6(1) may now be rearranged and written i n the form 6(2) where X aX1 - bX 2 - cX 5 = d *1 C D A P " f2 Sp Z2 X3 = (x^ + a = $AP = f1?a3 Va 2Dg 103 P m = 1 2„ 2<d . r I f 3 i n o <m s i n Qavm° va3c Scm^cos 2c><Jo + A? ( 2 xmV r @m> ( s r ) ) m = 1 = va5c S av L + A P J' \ sino^  (xm+1= xm) p - x/0 d i • 1 + > r s i a ^ c o s 9 a v m (9m - AP (2xm9m + r 9 m 2 ) _ va3cScm ~ <• ^ m o 1 + S H V a 5 c (1 + A P ) S n P 3 Equation 6(2) i s a linear equation i n three unknowns, thus satisfying the requirements of the proposed method of data analysis,, Now, from the elementary gas laws, several of the quanti-ties i n equation 6(2) may he written in terms of the basic system variables, temperature and pressure„ Therefore ?a3 a h _ 6(3) T3R 104 Taking into account feeder air losses, and Wac = » W a 6(4) v a3c " W a c R T 5 6(5) P A 3 D In P 3/P 2 1 In ( T 2 1 P^) ( T3 P21 ) 6(6) 17 The Sutherland formula ' permits calculation of the average viscosity of the a i r during transit in the duct 1555 T + C) ( A a v ) : v * 6(7a) 555 T~ w + C ) ( 7 7 ) av ' v T where/t7' » 0.18 @ T* - 0°F, and Tav ° T5° + T21 6(7b) 2 i where T^ i s an approximation of T at the center of the injector. Let T^' = = T j 0 + T ^ Q , where T j Q and T^ Q are the free running temperatures at points 3 and 4. The pressure drop over the duct i s AP = Pj = P 2 1 6(8) Fi n a l l y , the average Reynolds Numbers with respect to the duct and the particles are, respectively, R e = D Yaavfaav 6(9) 105 E e p = d ( v a a v " Y m a v ^ a a v 6(10) where v__,_ = v - . (1+ _£P )(as i n the i n t e g r a t i o n f o r 6(11) aav a:>c equation 3 ( 5 0 ) ) vmav " 2 4 0 Vm 5(1?) 6(12) F i n a l l y , the average s l i p r a t i o determined i n terms of the measured q u a n t i t i e s may he obtained from equations 6(11) and 5 ( 1 7 ) . S a v = vmav 6(13) v aav The c a l c u l a t e d average value of the s l i p r a t i o may be obtained from equation 3 ( 3 8 ) . Recognizing the f a c t that the quantities r , L, D, xm+1, x 0 A cos>3<. sin<»< sinQ -, cos9 ~. m, g, D, a v , p, m, ih, a v , and av are f i x e d q u a ntities r e l a t i n g the geometric configuration of the duct with respect to the earth, the above equations, together with those given i n Chapters 4 and 5, permit the computation of a l l the q u a n t i t i e s i n equation 6 ( 2 ), provided the appropriate v a r i a b l e s are measured during the test-runs. A summary of the quan t i t i e s involved i n equation 6 ( 2 ) , the numbers of the formulas used i n computing these q u a n t i t i e s , and the va r i a b l e s measured during the test-runs i s given i n Table 7. I t i s c l e a r from 106 Table V that many of the variables to be measured were re-quired more thanv;once. Eighteen separate readings were re-quired i n Table V. They are P 1 , P 3 , P4, P21 , A P1-3y ,^4-5, P a t , T3, T30, T40, T2T, T a t ; N b , Vhmx, ?b, *e, W s , and d. These readings constitute the minimum number required by the proposed method of data analysis. However, verification of the assumption that the pressure gradient i s essentially linear over the duct length required that pressure readings be taken at each duct tap, at least i n one test-run; this also permitted close examination of the pressure gradient at the elbows. Moreover, i n addition to readings taken with two-phase flow, that i s , with material being injected, i t was considered advantageous to have data on the flow characteristics for a i r alone flowing i n the duct at conditions similar to those during two-phase flow. Toward this end, readings were taken for the so-called "free-running" condition, for each blower speed, prior to each series of two-phase runs. Consequently, a single set of data on free-running conditions was obtained for each five sets of two-phase flow data. The blower speed, and, hence the range of ai r flow rates, was different for each six sets of data 0 Twelve such sets of data were obtained during each series of test-runs. The independent variable i n a l l of the test-runs was the material flow rate, Vm. In a l l , three series of test-runs were conducted, two for plastic pellets and one for barley. A complete set of 107 TABLE V SUMMARY OF TEST^ -RUN VARIABLES MEASURED FOR COMPUTATIONS Quantities to be Computed Formula Measurements Primary Secondary Number Taken AP 6(8) P39 P21 ?a 3 6(3) T 3 va3c 6(5) _ w a 5(4) pae V V p i t 5(8) P"P ^ Z L J ^ I ^ J V a c 6(4) m 5(15) Vhmx> ^b» *e S e m sm 3(34) -S„ m cm CD 3(37) 3(39) — 8ep 6(10) d A 6(7) T21» T 4 0 » T30 ^aav 6(12) V aav 6(11) • = n 6(5) V mav 5(17) s Sav 6(13) -Savc 3(39) f 1 3(45) 6(9) 108 pressure readings was taken during one of the series of test-runs conducted with plastic pellets. The other two series of test-runs were limited to measurement of the variables l i s t e d i n Table V, plus several pressures along the duct. A complete F F presentation of these data i s provided i n Tables 1, 2 and F 3 of Appendix I. SEQUENCE OF OPERATIONS The sequence of apparatus adjustments was designed to take into account the fact that one person was to take the readings, and with a view to making each cycle as short as possible. The readings were recorded i n their natural sequence. At the beginning of each test run,the blower variable speed drive was adjusted to provide the desired air flow rate, i n accordance with Figures 10 and 11 0 Both blower and injector motors were then started. The injector variable speed drive was adjusted to provide about 50 to 100 rpm at the injector r o l l s . Without material being injected into the duct, the blower was allowed to warm up u n t i l the temperature readings at points 3» 4- and 21 became stable. These temperature readings were then quickly recorded. The pressures at points 1, 3? 4-, 5 9 9» 12, 18 and 21 were recorded with reference to atmospheric pressure, using a mercury manometer and appropriate manipula-tions of the cocks i n the manometer assembly. In the f i r s t 109 test run with plastic pellets, the pressure readings at the remaining taps were recorded, with reference to the nearest preceding pressure at any of points 4, 9» 12 or 18, using a water manometer. These readings covered the free-running condition. The deflector at the top of the collector chute was positioned such that particulate material would he returned from the receiver to the injector and thus he circulated con-tinuously. With the injector r o l l s s t i l l turning at from 50 to 100 rpm, particulate material was introduced into the collector chute through i t s hinged door. Sufficient material was placed into the chute to create a head of about 10 to 15 inches and a volume of 400 to 700 cubic inches over the injector r o l l s . Since the system reached steady-state conditions within seconds, i t was possible to begin taking readings immediately. The sequence of temperature and pressure readings was the same as that given above for the free-running conditions. After the temperature and pressure readings were taken and recorded, attention was turned to the injection characteristics, V t W as measured by hmx' e' and s. The weight of the material i n the duct under steady-state conditions;; was determined "by turning off the injector drive and repositioning the collector chute deflector simultaneously so that the material in the duct was collected i n the weigh-barrel. The injector seals acted as brakes on the injector r o l l s , thus stopping the r o l l s almost 110 simultaneously. The strain indicator reading from the weigh-barrel being recorded, the material 'in the barrel was returned to the collector chute. The collector chute deflector was returned to the v e r t i c a l position. The injector variable drive required that, ideally, i t should be started only under unloaded conditions. Consequently, i t was necessary to note carefully the position of the variable speed adjustment handle, return i t to dead center, restart the injector motor, and return the handle to i t s original steady-state position, and thereby return the injector r o l l e r s to their original speed. With the system returned to steady-state conditions, the V hopper volume over the injector, hmx, was recorded. The collector chute was then repositioned for discharge of the particulate material into the weigh-barrel; simultaneously a stop-watch was started to obtain the time, *e, required to empty the hopper from steady-state conditions. These readings com-pleted one set of data. The injector r o l l e r speed was now increased s l i g h t l y and the entire cycle repeated for the increased W particle feed rate, m. As mentioned previously, five different W values of m were obtained for each blower speed. SUMMARY The experimental procedure incorporated i n this research has been b r i e f l y outlined i n this chapter. The reasons behind 111 the procedure, such as the proposed data analysis and the limitations of a one-person operation, have been discussed. The system variables, their selection, and the sequence and method involved i n their measurement have also been adequately covered. CHAPTER 7 ANALYSIS OF EXPERIMENTAL RESULTS The proposed procedure for taking the experimental results, as outlined i n Chapter 6, was closely followed. I t was originally intended that a single series of ten test-runs would be conducted for each of the two particulate materials used, namely cellulose pellets and barley grains. However, the f i r s t series, with cellulose pellets, was plagued with procedural d i f f i c u l t i e s , such as sticking i n the injector seals and excessive time lapses during pressure readings, thus giving r i s e to experimental error and inconsistencies i n the subse-quent"analysis. Therefore, an additional series of test-runs was conducted under somewhat more favorable circumstances. A l l three sets of data are presented i n Appendix I of this thesis. A detailed discussion of the tabulated presentation of the experimental data w i l l be forthcoming during the consideration of the various topics included i n this chapter. The f i r s t topic to be discussed i s that of Pressure Gradients, insofar as they relate to the overall computation as outlined i n Chapter 6. Attention i s then turned to the computation of flow properties, s l i p ratios, ratios of polytropic gas constants, and the solution of the simultaneous equations obtained from equation 6(2). wherever practicable, examples of the experimental and calcu-lated data are presented graphically throughout the text. PRESSURE GRADIENTS The experimental results and the calculated values of flow properties are presented i n Appendix I. The series of test-runs are numbered 1, 2 and 3» for the cellulose pellets, barley and cellulose pellets, respectively.. Tables F1, F2 and P3 present the tabulated values of the quantities required by the procedure outlined i n Chapter 6« Test-runs 1 and 7 i n each case were conducted with the injector running but with no particulate material being injected into the air stream. It was found that two separate blower speeds provid-ed an adequate range of values of the dependent variable, y y W ac/ m. At each blower speed, of course, m was the independent variable, being increased from low to high values, and being determined from ^hmx, W s and *e. Several additional pressure readings along the duct, at points 9» 12 and 18, were taken to provide an indication of the pressure gradient along the duct. Table G contains a complete set of pressure readings, at a l l the pressure taps along the duct, for a l l the test-runs i n series No.1. In relating the pressure readings to the duct and tap locations, i t may be helpful to the reader to refer to 114 Figure 9? which shows the system schematic An example of the pressure gradient along the duct i s plotted in Figure 22. The equivalent length values plotted PRESSURE GRADIENTS CELLULOSE PELLETS Fig„22 Approximate Linearity of Pressure Gradients. along the abscissa were taken from Table B of Appendix I, with 0 representing the centerline of the injector. The pressure at the centerline of the injector i s taken to be that at point 3, which i s only about 16 inches upstream from that point. 115 The value of this pressure reading, as well as those at points 4, 9, 12, 18 and 21 were taken from test-runs 7 and 12 of test-run series No.5, as presented i n Table F3 of Appendix I. L= E.<OlHV. L E . N G T H , fh Fig . 2 3 Effects of Non-Positive Displacement Blower. The almost exact l i n e a r i t y of the pressure gradient for free-running conditions i s self-evident. However, the gradient for two-phase flow i s substantially non-linear. The approximation 116 used i n the theoretical simplifications presented i n Chapter 3 i s indicated by a broken l i n e . Clearly, there must exist a li m i t where the approximation of l i n e a r i t y of pressure gradient becomes inadequate. The sharp drop i n pressure across the injector i s accounted for partly by the constriction i n the duct which the injector represents and partly by the a i r loss at the injector. Under two-phase flow conditions, the sharp pressure drop i s no doubt largely accounted for by the almost instantaneous acceleration of the particles at the duct, although the factors applying to free-running conditions s t i l l contribute to this sharp decrease i n pressure. Another example of pressure gradient, this time for barley grains, i s presented i n Figure 23. The data were taken from Tables A and F2 of Appendix I, similarly to those for Figure 22. The pressure readings are for test-runs 7 and 12 of test-run series No.2. The consequences of the use of a non-positive displacement blower are readily i l l u s t r a t e d by Figure 23. Although the free-running pressure i s high and the i n i t i a l velocity i s that deter-W mined by the free-running value of aco, the injection of the second phase increases the a i r pressure, thereby substantially reducing the air flow rate and the i n i t i a l a i r velocity. A momentary increase i n pressure, due to slugging or a sudden 117 change i n the feed rate can reduce the a i r flow rate below that required to prevent s a l t a t i o n . Another f a c t o r which contributes to the drop i n the i n i t i a l pressure, due to f l u i d flow alone, i s the energy removed from the f l u i d by the ac c e l e r a t i n g p a r t i c u l a t e phase and the attend-ant decrease i n f l u i d s p e c i f i c volume and v e l o c i t y . Figure 23 also bears upon the approach and experimental work of many of the writers and researchers reviewed i n Chapter 2, Almost i n v a r i a b l y the pressure drop during two-phase flow i s taken to be that occurring under free-running conditions plus a component due to the presence of the second phase. Nowhere i n the l i t e r a t u r e was mention found of the change i n the flow conditions of the a i r which attends the increase i n pressure and the i n j e c t i o n of the second phase. This, of course, does not cast conclusive doubt on a l l the work done by others, but f o r c l a r i t y , a d i s t i n c t i o n should be made between the pressure drop due to the flow of the single f l u i d phase under free-running conditions and under two-phase flow conditions. The v a r i a t i o n of the pressure gradient i n the v i c i n i t y of elbows and v e r t i c a l sections of the duct may now be analysed. Figure 24 shows these v a r i a t i o n s , i n the f i r s t h a l f of the duct, f o r t e s t-run s e r i e s No .1, test-runs 1 and 6. The pressure readings were taken from Table G of Appendix I. I t i s note-worthy that the s l i g h t increase i n pressure which occurs at the 118 elbows occurs not only f o r two-phase flow, but also f o r f r e e -running conditions. These deviations appear to be e n t i r e l y l o c a l i n nature, i n s o f a r as the o v e r a l l l i n e a r i t y of the pressure gradient i s concerned. Another noteworthy conclusion PRESSURE GRADIENTS T E S T - R U N S E R I E S * 1 T E S T - R U N S \ 4 (s * = -794 oo . Her •< 7.> 9. io zo 3 o AO so 6o 70 L = E.QOIV. LEKIGTH* f f 8 0 90 / C O /io 126 I30 Fig.24 Pressure Gradient V a r i a t i o n s at Injec t o r , Elbows, and V e r t i c a l Sections of Duct. which may be derived from Figure 24 and the ad d i t i o n a l data contained i n Table G of Appendix I i s that the pressure gradient i s not s i g n i f i c a n t l y affected by v e r t i c a l flow under e i t h e r 119 single-or two-phase flow conditions. S t i l l another important conclusion which may be drawn from Figure 24 i s that the i n i t i a l pressure drop, due to the acceleration of the second phase, occurs i n the immediate v i c i n i t y of the injector, probably within two or three feet. Indeed, this conclusion was qualitatively v e r i f i e d visually i n the glass section of the duct downstream from the injector. No saltation occurred at pressure tap #4 i n the glass duct, as would have been the case had the material not already attained a high velocity. Moreover, the visual appearance of the par-t i c l e velocities at points 4 and 5 was perceptively the same. COMPUTATIONS The computations outlined i n Chapter 6 were exceedingly lengthy and tedious. The use of the University's IBM 7040 computer was therefore imperative. A program was written to carry out a l l of the arithmetic operations involved i n computing the air flow properties from equations 6(3) and 6(12>) and the other equations referred to i n Table V of Chapter 6. The co-efficients i n equation 6(2) then having been calculated for each test-run where ^m^, each of the 120 combinations of three equations was solved simultaneously. The output of the computer program was too bulky to permit the inclusion of a sample i n this thesis. However, i n the interest of future research and brevity i n this section of the thesis, the source 120 program, i n Fortran IV language, i s included i n Appendix II. The many comment statements throughout the program should make i t readable to anyone having even a slight knowledge of Fortran IV language. As a further aid, the program i s prefaced with a general outline relating the equations summarized i n Table V to the program. Air Properties and Flow Characteristics Tables H1, H2 and H3 contain the computed values of the air properties and flow characteristics with respect to the duct and particles. Over the entire series of tests, values of the a i r density at the injector, a^3» vary from 0 .0815 to 0 .0904 l b / f t ^ ; average values of a i r density during the flow process range from 0.0770 to 0 .0815 l b / f t ^ . Viscosity varies from 0 .386 x 1 0 ~ 6 to 0.411 x 1 0 ~ 6 l b - s e c / f t 2 . The Reynolds number for the a i r flowing with respect to the duct varies from 0 . 4 6 9 x 10^ to 1 .097 x 10^; with respect to the particles, from 0 .106 x 10^ to 0 .402 x 1 o \ provided the i n i t i a l short period of particle acceleration i s ignored. The drag coefficient of the particles, assuming that irrespective of shape,,a tumb-li n g particle i s equivalent to a sphere, varies from 0 .355 to 0 . 4 1 3 . f The Darcy-Weisbach f r i c t i o n factor, 1, for flow in the duct, varies from 0 .0272 to 0 .0386. It should be noted that for a 2-inch standard pipe duct made of clean steel, the f r i c t i o n 121 factor would normally be between 0o022 and 0.028, approxi-mately, depending on the Reynolds number for the flow of a i r in the duct. In fact, these values were obtained experiment-a l l y i n the determination of Injector Air Losses. However, these values were too low to satisfy the free-running condi-tions, after the preliminary tests had been conducted with cellulose pellets, vetch seed, Australian Winter Peas and other materials. It i s known that, prior to the preliminary tests, the duct had not been used to any extent and could be assumed to have had an interior surface with a hydraulic roughness resembling that for clean steel. However, after the preliminary tests had been conducted, i t was noted that the interior of the glass section of the duct had become coated with a layer of fine dust. This might have been caused i n the following manners the cellulose pellets had been contaminated with o i l during an earlier research and, hence, had a s l i g h t l y o i l y surface prior to the preliminary test-runs. In addition, several of the seeds used i n prelimin-ary runs contained o i l which they might have given up during crushing or shattering i n the duct or during pressing between tJa«-»JLnjector r o l l e r s . F i n a l l y , the preliminary test runs which were concerned with the injector a i r losses entailed the use of o i l at the injector seals, conceivably some of this o i l was carried into the duct by the a i r stream. Subsequent tests with materials such as r i c e , which suffered a high degree of particle 122 damage and created considerable dust, coated the o i l y duct surface with a layer of fine dust. This explanation seems to be borne out by the fact that the f r i c t i o n factors for the second run on cellulose pellets, after the barley tests, i n which dust was produced, were considerably higher than those obtained during the f i r s t pellet test-run. f The increase i n 1, which was noted on the basis of sat i s -fying the free-running conditions and assuming that the previous evaluation of injector a i r losses and blower discharge rates were correct, entailed an increase i n the duct surface relative 17 roughness ratio ' from 0.00015 to 0.00060. This increase seems entirely plausible i n view of the appearance of the inside of the glass portion of the duct. The polytropic gas constants were found to vary from 1.4-7 to 1.98. Values of the ratio of the polytropic gas constant with two-phase flow to that with single-phase f l u i d flow are tabulated as R1 i n Table M1, M2 and M3. Figure 25 shows a plot of n/n Q and n versus W a c/V m« The values of n and n/n Q were plotted against several other para-meters (R^, Hg, etc. i n Tables M1, M2 and M3) but the results were equivalent to those i n Figure 25. From the appearance of Figure 25, i t might be concluded that the flow process for a i r , during two-phase flow, approaches an irreversible adiabatic process when the air-to-material ratio approaches zero. 123 On the other hand, i t must he remembered that the values of n and n Q were obtained from a closed duct i n s i d e a b u i l d i n g where the temperature outside the duct was always lower than that i n s i d e , even at pressure tap 21. Consequently, the poly-t r o p i c gas constant was always greater than that which would 2.0 o< • ( 1 ) m 0 0 n O a o o ^ Q u o I X - _ o f \> o Tr -' o \ u O 1 E S T - E R I E S * 1 P E i LETS • 1 E S T - ?UM S E R I E 5 **2 P E L I E T S 0^ E S T 1 E R I E S # 3 P E L i .ers o cc p Z o a o a P o a .2. .3 -4 - 5 .6 A I R - T O - M A T E R I A L RATIO, W»c/V\Ar, • 8 l.O Pig.25 C o r r e l a t i o n of Polytro p i c Gas Constant with A i r to Material Ratio. be expected f o r an i r r e v e r s i b l e adiabatic process but appropriate f o r one i n which heat was transf e r r e d from the duct to the duct environment. In the case of a duct located, say out-of-doors, where the extremes of summer and winter weather conditions are such that heat would be tra n s f e r r e d into the duct i n the summer 124 i ! and out of the duct i n the winter, the polytropic gas constant, for low values of the air-to-material ratio would he respect-ively less than and greater than that for an irreversible adiabatic process. Solution of the Three Sets of Ten Simultaneous Equations for f2» f j and f^. The three sets of 10 simultaneous equations resulting from the three series of test-runs were solved three at a time, as planned i n Chapter 6. The computer routine, beginning with statement No.250 of the program i n Appendix.II, was designed to select a l l possible combinations of three equations, of which 120 such combinations exist for 10 equations i n three unknowns. Each series of test-runs was dealt with in turn by the computer. After many computer runs, during which time the program was thoroughly debugged, a l l possible sources of experimental error were checked out, and the mathematics i n Chapters 3 and 6 was thoroughly checked, i t was found that, for each of the three series of test-runs, the 120 values of each of the f r i c t i o n factors, f2? f j and f^, varied considerably and were predomin-antly negative. The significance of this development w i l l be considered presently. The values of f^ and f ^ were specified at the beginning of the program to permit the calculation of Sg, Sy, S^ . and S a v as outlined i n statements 113 to 14-3. Of course, this procedure 125 assumed the constancy of f^ and f^ over the range of ^ & c / ^ m i n the tests conducted. It was hoped, o r i g i n a l l y that the solutions to the simultaneous equations would produce reason-able values of f2» f^ and f^, such that an i terat ive proced-ure could be used to establish the i n i t i a l values of f^ and f^ i n the determination of the s l i p ra t ios for the coeffic ients of equation 6(2). Unfortunately, this was not possible because the obtained values of f^ and f^ were negative. Consequently the values of f , and L were varied systematically to see i f there was a consistent variat ion of -s-s^ with some system r 2 b p parameter such as the air - to-material r a t i o : f^ was varied from 0 .005 to 1.50 and f . was varied from 0.000 to 0.020. °D An At the same time, the value of ^ 0 r was calculated i n each f 2 S p case by statement No.240, assuming f^ and f^ to be held constant, such that the coeff ic ients of equation 6(2) were s a t i s f i e d . Values of JL?, S H , S V , S E and S avc for a sample of the calcu-2bp lations carried out under the above procedure are tabulated i n Tables K1 to K10 of Appendix I I ; only those which seem i n reasonable agreement with the experimental results are included. Very low values of the s l i d i n g f r i c t i o n c o e f f i c i e n t , f^, produce a very high average s l i p r a t i o , S Q T r „ , and high values of the CDA kinetic f r i c t i o n factor , f^, result i n negative values of £ 2 p °D Ao The tabulated values of - ^ c lear ly indicate that f0 must 2bp * vary with some system parameter, perhaps the air - to-material ra t io or the reciprocal of that r a t i o . But i f f2 varies with 126 the a i r - t o - m a t e r i a l r a t i o , i t seems reasonable to expect D to vary a l s o , since f 2 and C D are d i r e c t l y r e l a t e d to the flow of f l u i d around the p a r t i c l e s . On the other hand, i f f P and CJJ vary at the same r a t e , then the values of * i n Tables K1 to K10 would be constant. They are not constant. Therefore and ±2 fflust vary independently with some system parameter. Among the many parameters with which f 2 and might vary, the flow rate parameters included i n Tables M1, M2 and M3 are possible choices. Those contained i n Table N, which were 14-studied by Zenz , also were considered. Of these parameters, the simplest and most useful r e s u l t s were obtained with the simple a i r - t o - m a t e r i a l r a t i o and i t s r e c i p r o c a l , as tabulated i n Tables J 1 , J2 and J 3 . The consideration of fo» as i t appears i n the parameter .» q • and the r e c i p r o c a l of t h i s parameter, r 2 b p and the consideration of Cj as i t appears i n the calculated s l i p r a t i o s , are presented i n the next two sections of t h i s chapter. V a r i a t i o n of f 2 with A i r - t o - H a t e r i a l Ratios C DA The values of ~ 0 " f o r various combinations of f 2 and f,. f 2 S p 5 4 are tabulated, as mentioned, i n Tables K1 to K10 of Appendix I . The values f o r the f i r s t s e r ies of test-runs are not consistent with each other or with those of the second and t h i r d s e r i e s , due, undoubtedly, to the procedural d i f f i c u l t i e s mentioned 127 e a r l i e r . These results are therefore not plotted i n the following graphs. C DA Figure 26 shows a Plot of >. g ? versus V L /W » Since 1 f a c m 2 p Ap/Sp i s a constant related to the cross-sectional and surface areas of the particles, depending only on particle shape and being equal to 0.25 for a sphere, i t may be 0 P e u • BARf .ETS E Y 0/ /A o V rn ° 0 Off* LJ 0 >0£7S • A ^POINTS > D •2 - 3 4- . 5 -6 -7 A I R - T O - M A T E R I A L - RATIO, W«c/W«" . 9 l-o Fig.26 Variation of Cp/f- with W" A L for Various Combinations of f 5 and f^. * ac m assumed that the dependent variable i n Figure 26 i s CD/f2<> The curves i n Figure 26 show that for very low values of f^, the ratio of C D / f 2 i s nearly constant. As the value of f ^ increases, and f^ decreases, the ratio increases sharply with 128 W /VT ac m* ° D A T > Figure 27 shows a plot of ^  S P versus W^W^ for one set 2 p of the values of f , and f,, used i n Figure 26* This plot i n -f 2 S T > dicates that the value of H a p approaches a maximum value at D T ) high values of w * m / W a c < , This i s what would he expected, inasmuch as an increase i n W 7w represents an increase i n the number m ac . of particles around which the a i r must pass during transit of f2;\> the dueto It may be that the value of * * could be expressed CD Sp i n terms of an exponential function of the following form: _ ™ f+ e=K2 J L ) . 7(1) where and K 2 depend on the particle properties and the values of £j and If Sp/^j, i s included i n K^,equation 7(1) becomess f 2 . ^ C j , (1 - e"* 2 V ) 7(2) Vac I t should be noted, now, that the two particulate materials represented i n Figures 26 and 27, namely barley and cellulose p e l l e t s , were very different i n shape, size and surface rough-ness 0 The barley grains had a mean diameter of 0.120 inches and the cellulose pellets a mean diameter of 0.160 inches. The barley grains were e l l i p s o i d a l i n shape; the cellulose pellets were c y l i n d r i c a l i n shape with the diameter equal to the 129 length. The c e l l u l o s e p e l l e t s were very smooth; the barley grains had a porous surface. But, even so, there i s a remark-able c o r r e l a t i o n between the data from both materials. This s i m i l a r i t y may be accounted f o r by the f a c t that, on the basis of equation 4(3)b, from which the f r e e - f a l l i n g Ck. t^f u D B A 0 PE * L E Y - L E T S 0-20 0 .O027 s a. A o a —"V* o N ^ A T E R V A L - T O - A > R R A T ' I O \ A M / W a t c P i g 0 27 V a r i a t i o n s of f~/C with W AT f o r One Combination of f5 and f 4 , d m a c v e l o c i t y i s V 5 C D Pa the f r e e - f a l l i n g v e l o c i t y f o r both p a r t i c l e s , assuming a sp h e r i c a l shape, i s approximately the same. That f o r the barley 130 grains i s 32.9 and that f o r the c e l l u l o s e p e l l e t s i s 34.1 f t / s e c . In other words, the increase i n density of the b a r l e y grains over the c e l l u l o s e p e l l e t s i s o f f s e t by the decrease i n p a r t i c l e diameter. A l l t h i s seems to point to the conclusion that as i n the work of Richardson and McLeman and Kikkawa et a l , the s i z e , shape and surface c h a r a c t e r i s t i c s of a given type of p a r t i c l e are taken in t o account by the f r e e -f a l l i n g v e l o c i t y of the p a r t i c l e s . Insofar as t h i s t h e s i s i s concerned, no attempt was made to determine the constants and K 2 i n equation 7(2) f o r any materials. Nor was an attempt made to determine values of actual f r e e - f a l l i n g v e l o c i t i e s f o r the materials. Such work would necessitate the assembly of new and s l i g h t l y more sophis-t i c a t e d apparatus than that used i n the present research and was therefore beyond the scope of the research. Comments and recommendations regarding the determination of these quantities are contained under Recommendations f o r Future Research i n Chapter 8. S l i p Ratios and Evaluation of C D The experimental s l i p r a t i o data and values of S a v are contained i n Tables J 1 , J2 and J3. The c a l c u l a t e d values of Sjj, Sy, S E and S a v c f o r various combinations of f ^ and f ^ , as pr e v i o u s l y noted, are contained i n Tables E1 to K1G. For 131 purposes of comparison with Figure 8 i n Chapter 4, Figure 29 il l u s t r a t e s the variation of the average material velocity with the average air velocity over the f u l l length of the duct. The points in Figure 28 represent the experimentally measured values of vaav and vmav. The curves drawn through the points represent the average values of slip-ratio calcu-lated for various combinations of f^ and f ^ and applied to the appropriate values of vaav for the various sub-test runs. The appearance of Figure 28 indicates that the calculated S values of the average s l i p r a t i o , avc, do not entirely conform with the experimental values. This i s not entirely surprising, i n view of the fact that, as previously noted, f 2 varies with the air-to-material ratio i f f ^ and f ^ are constant, thus suggesting that C D may also vary. The dependence of the s l i p ratios, for curved and straight ducts, on the particle drag co-eff i c i e n t are readily discernable upon the examination of equations 3(34) and 3(37)» An increase i n the drag coefficient increases the values of and B 2 i n equation 3(34), thus i n -creasing the value of the s l i p r a t i o . Similarly i n equation 3(37)9 an increase i n Cp increases the s l i p r a t i o . On s t r i c t l y i ntuitive grounds, i t i s obvious that an increase i n C^ w i l l increase particle velocity and, hence, the value of the s l i p ratio. Moreover, i t i s quite reasonable to expect the drag coefficient to increase, as the number of particles in the particulate phase increases, because the matrix of particles 132 through which the a i r must flow becomes more dense. But, whereas the relative flow f r i c t i o n factor, f 2 , appears to i n -crease from zero to some constant value for large values of W /W" , i t would he expected that would increase from i t s m ac' * B value for f r e e - f a l l i n g particles instead of from zero, although o 70 O \o 20 30 40 50" 55 TO So 9 0 100 no A V E R A G E A I R V E L O C I T Y > \ & « v . j Fig.28 Experimental Values of v„„„„. and v^^„ a&\ mav probably i n an exponential fashion similar to that of f 2 . A consideration of the variation of i s f a c i l i t a t e d by Figure 2 9 , whez'e the average values of the calculated and experimental s l i p ratios are plotted against the air-to-133 material r a t i o . As before, the values of S„ n are taken f o r ' avc various combinations of f ^ and f ^ . Figure 30, i s s i m i l a r to Figure 2 9 , but values of S a„ and S_„„ are p l o t t e d against 9.V 3.VC m a t e r i a l - t o - a i r r a t i o s , W /VL , The decrease i n the ca l c u -* m ac la t e d values of S a v c i s due to the decrease of a i r v e l o c i t y i n the experimental test-runs, when the m a t e r i a l - t o - a i r r a t i o o 1 a _i m < or •7 •o .5 4 -3 -2 0 \ V * -'ELLE.T PARLEY > k o V \ -p f 5 [ 1 D<>[ P 0 - o o -2 -3 «4 -5" -6 -7 -8 A I R - T O - M A T E R I A L . R A T S O W a c / W m F i g . 2 9 Average S l i p Ratios Versus A i r - t o - M a t e r i a l Ratio was increased. Equations 3 ( 3 4 ) , 3 ( 3 7 ) and 3 ( 3 8 ) are such that f o r a constant aav the value of a.vc would remain constant as shown i n Figure 30. On the other hand, as i n the case of the values of f 2 i n Figure 2 7 , the experimental data i n Figure 30 in d i c a t e s that the drag c o e f f i c i e n t achieves a maximum, r e l a t i v e l y stable value f o r high values of W /Won. in sic 134 The form of the drag c o e f f i c i e n t f u n c t i o n may he s i m i l a r t o t h a t of f 2 , say, GJ m GN U + K x (1 D " - D ( ' * ^3 v ' " ^ V ] W a c where C D i s the drag c o e f f i c i e n t of a s i n g l e f r e e - f a l l i n g p a r t i c l e . O PE O B/ LLETS LLLETS • • a • -> ^ n < > 6 0 < to UJ O < a. in MATERIAL-TO-AiR RATIO > ^ / W o c Fig.30 Average S l i p R a t i o s Versus M a t e r i a l - t o - A i r Ratios I t i s worth summarizing at t h i s p o i n t what seems to he the r e l a t i o n s h i p between f 2 , C D, and Wm/w*ac<> F i g u r e 31 shows a s e r i e s of curves r e l a t i n g these f a c t o r s . The p a r t i c l e c h a r a c t e r i s t i c s , C-p. and f 0 i n c r e a s e w i t h V /W 7 JJ d m ac 135 f 2 increases more r a p i d l y than C-^ because the l a t t e r has i t s f r e e - f a l l i n g value at ^ /W = 0, whereas f 2 i s equal to 0 at W\A/ = Oo The curve f o / 0 n i s e s s e n t i a l l y that obtained i n m ac d D * Figure 26« The values of C-p and f 2 both a t t a i n a maximum value f o r high values of W /W„ » I t seems possib l e that °^ m ac H Y P O T H E T I C A L V A R I A T I O N S OF f 2 AND C D r\F * V C D IS CALCULATED ON THE BASIS OF PRESSURE. LOSSES, I r s MAyiMUM \ VALUE D E P E N D S OM f j AND — u * M I N I U M VALUE OF C D IS T H A T O F A SIMGLE PART\CLE D U LL O ui r> M A T E R I A L - T O - A I R R A T I O WVM/W oc F i g o 3 1 Hypothetical Variations of f 2 and with M a t e r i a l - t o - A i r Ratio. high values of Wm/Wo/1 maj represent saturated flow i n which in 9LC the flow c h a r a c t e r i s t i c s of the two-phase mixture s t a b i l i z e and the mixture behaves as a single fluid„ 136 SUMMARY This chapter has dealt with pressure gradients during single-phase flow with a i r and two-phase flow with a i r and s o l i d p a r t i c l e s ; the dependence of the p o l y t r o p i c gas constant, f o r the flow process, on the m a t e r i a l - t o - a i r r a t i o ; the evaluation of the a i r properties during two-phase flow; an attempt, based on t h e i r presumed constancy, to determine the values of f 2 , f j and f ^ f o r two-phase flow; and an hypothesis regarding the v a r i a t i o n of C D and f 2 i f f ^ and f ^ are constant. The conclusions and recommendations which may be drawn from t h i s chapter, as well as those pre-ceding i t , are presented i n the next and f i n a l chapter of t h i s t h e s i s . CHAPTER 8 CLOSURE This t h e s i s i s drawn to a close by means of t h i s f i n a l chapter. The preceding seven chapters are r e c a p i t u l a t e d , conclusions are stated, and recommendations f o r future research are o u t l i n e d . RECAPITULATION The preceding seven chapters of t h i s t h e s i s have dealt with the various aspects of a research project which was designed to throw some l i g h t upon the phenomenon of two-phase flow. Two-phase flow was shown, i n the Introduction, to be a natural phenomenon. I t s e a r l y and present day a p p l i c a t i o n to engineering problems, on the basis of often dubious empirical methods and equations, was discussed. The problem was stated, i n s o f a r as t h i s t h e s i s was concerned, to be p r i m a r i l y a matter of r a t i o n a l i z i n g the multitude of v a r i a b l e s which a f f e c t two-phase flow, p a r t i c u l a r l y pneumatic conveying. A l i t e r a t u r e search was conducted with a view to deter-mining whether previous researchers had s u c c e s s f u l l y r a t i o n -a l i z e d these v a r i a b l e s . A l l but one paper, that by Kikkawa 1°/ et a l , were based almost e n t i r e l y on dimensional analysis 138 and other empirical methods; the paper by Kikkawa was discovered i n the f i n a l stages of the composition of t h i s t h e s i s . Upon completion of the l i t e r a t u r e review, a theory was postulated, wherein the flow of a p a r t i c l e suspension was considered to be analogous to the flow of a homogeneous f l u i d . This theory was developed to cover a closed duct of any si z e and configuration, made up of curved and s t r a i g h t sections, with a two-phase mixture c o n s i s t i n g of a compressible f l u i d and s o l i d non-cohesive p a r t i c l e s of any shape, s i z e , and other such c h a r a c t e r i s t i c s , flowing within the duct. The theory, as i t applied to pneumatic conveying was then evaluated i n the l i g h t of the experimental r e s u l t s contained i n several of the pub l i c a t i o n s reviewed e a r l i e r . An apparatus was designed and developed f o r v e r i f y i n g or r e f u t i n g the theory. The apparatus consisted, e s s e n t i a l l y , of the basic components which make up the pneumatic conveyors presently used i n industry: a blower, i n j e c t o r , duct with v e r t i c a l , h o r i z o n t a l , and elbow sections, and a cyclone r e c e i v e r . The experimental procedure was designed to f a c i l i -tate the determination of the three f r i c t i o n f a c t o r s which were inherent i n the t h e o r e t i c a l development, provided these f a c t o r s were constant over a range of flow conditions. Other-wise the r e s u l t i n g data were to be analyzed i n an attempt to disce r n the nature of the v a r i a t i o n of the f r i c t i o n f a c t o r s . The experimental r e s u l t s were analyzed i n accordance with 139 the proposed procedure with the a i d of an e l e c t r o n i c computer. As much information as possible was gleaned from these r e s u l t s , CONCLUSIONS The l i t e r a t u r e review contained i n t h i s t h e s i s , while not a l l - i n c l u s i v e , revealed that the present state of under-standing i n the f i e l d of two-phase flow, as i t i s applied to mining engineering, chemical engineering, and even the trans-p o r t a t i o n of capsules i n transcontinental p i p e l i n e s , i s very l i m i t e d indeed. There e x i s t s a growing need f o r the r a t i o n a l -i z a t i o n of t h i s natural phenomenon. I t may therefore be con-cluded e n t h u s i a s t i c a l l y that t h i s f i e l d i s one of the most challenging i n applied research today. A p p l i c a t i o n of the t h e o r e t i c a l analysis presented i n Chapter 3 to the experimental data contained i n several of the f i e l d p u b l i c a t i o n s showed that the t h e o r e t i c a l analysis has considerable merit. Therefore i t may be concluded that the experimental work described i n t h i s t h e s i s was warranted. A new type of i n j e c t o r was developed during the course of t h i s research. Tests and developmental work on the appara-tus showed that i t s main advantage l a y i n i t s steady, c o n t r o l l e d i n j e c t i o n r a t e . This advantage was s l i g h t l y o f f s e t by the i n -adequacy of the seals at the ends of the i n j e c t o r enclosure and by the tendency of the apparatus to eject a large proportion of 140 c e r t a i n i n j e c t e d materials. However, i t may "be concluded that the a p p l i c a t i o n of a small amount of ingenuity and engineering s k i l l , towards overcoming these d i f f i c u l t i e s , may well render the apparatus extremely use f u l i n further research i n the f i e l d , as well as i n an i n d u s t r i a l context. I t was shown that the pressure gradient along a duct during single-phase flow of a compressible f l u i d , such as a i r , was very nearly l i n e a r . In a d d i t i o n , i t was shown that the pressure gradient f o r two-phase flow.with a i r and s o l i d p a r t -i c l e s , while not l i n e a r , could be approximated as being l i n e a r i n the case of small pressure drops (say up to 10 inches of mercury) across the ends of the duct. This approximation permits the determination of the v e l o c i t y at any point i n the duct as a f u n c t i o n of the distance along the duct. However, f o r pressure drops l a r g e r than 10 inches of mercury, the r e l a t i o n s h i p involved should be examined f o r the e f f e c t s of the l i n e a r approximation and, i f p o s s i b l e , a better approxima-t i o n or an exact function should be devised or sought. The pressure drop due to the flow of a i r alone, under free-running conditions, was found to be considerably higher than that f o r the a i r alone under two-phase flow conditions. This e f f e c t was due p r i m a r i l y to the decrease i n the a i r flow-rate with r i s i n g pressure, as r e l a t e d to the q u a s i - p o s i t i v e -displacement blower discharge c h a r a c t e r i s t i c s , but p a r t l y due to the k i n e t i c and thermal energy t r a n s f e r from the a i r to the 141 p a r t i c u l a t e phase upon i n j e c t i o n of the p a r t i c l e s . There-f o r e , i t may he said that whereas most researchers i n t h i s f i e l d regard the pressure drop during two-phase flow as being comprised of two p a r t i a l pressure drops, one f o r the a i r alone and another f o r the p a r t i c u l a t e phase, a c l e a r d i s t i n c t i o n should be made between the p a r t i a l pressure drop f o r the a i r under two-phase flow conditions and the pressure drop due to the flow of a i r under free-running conditions. Pressure f l u c t u a t i o n s at elbows, v e r t i c a l sections, and the i n j e c t i o n area were examined. A sharp drop i n pressure occurs across the i n j e c t o r , p a r t l y due to a i r l o s s e s , p a r t l y due to the c o n s t r i c t i o n i n flow c r o s s - s e c t i o n created by the i n j e c t o r enclosure, but p r i m a r i l y due to the acc e l e r a t i o n of the p a r t i c l e s i n that area. The pressure gradient does not d i f f e r i n slope or curvature to any perceptible degree f o r v e r t i c a l and h o r i z o n t a l ducts. This observation no doubt applies equally as well to i n c l i n e d s t r a i g h t ducts. A curved duct such as an elbow appears to experience a minor r i s e i n pressure across i t s ends when i t i s located between two s t r a i g h t ducts. However, i t may be concluded that the f l u c t u a t i o n s i n the pressure gradient, which occur at elbows, are l o c a l and i n s i g n i f i c a n t . The pressure gradient i s e s s e n t i a l l y l i n e a r or smoothly curved over the entire length of a duct of any geo-metric configuration. The p o l y t r o p i c gas constant f o r the a i r was found to 142 decrease i n value as the m a t e r i a l - t o - a i r r a t i o i n c r e a s e d 0 I t s v a l u e approached t h a t of i r r e v e r s i b l e a d i a b a t i c expansion f o r v e r y h i g h m a t e r i a l - t o - a i r r a t i o s . However, i t was noted t h a t , depending on the c o n d i t i o n s p r e v a i l i n g as regards heat t r a n s f e r between the duct and the duct environment, t h i s constant might take on values g r e a t e r than or l e s s than t h a t f o r i r r e v e r s i b l e a d i a b a t i c expansion or than those f o r the c o n d i t i o n s which p r e v a i l e d d u r i n g t h i s r e s e a r c h . Since t h i s gas constant e n t e r s i n t o the t h e o r e t i c a l , a n a l y s i s presented i n t h i s t h e s i s , o n l y i n s o f a r as the v e l o c i t y f u n c t i o n i s concerned, i t would be p e r s p i c a c i o u s , i n the a p p l i c a t i o n of the t h e o r y , to choose the value of the p o l y t r o p i c constant such t h a t i t i s s m a l l e r than t h a t expected f o r the p r e v a i l i n g c o n d i t i o n s . A n a l y s i s of the data from three t e s t - r u n s , c o n s i s t i n g of twelve s u b - t e s t s each, showed t h a t e i t h e r one or a l l of the f r i c t i o n f a c t o r s £ 2 , f j and f^_, which are i n h e r e n t i n the t h e o r e t i c a l a n a l y s i s presented i n t h i s t h e s i s , may not be con-s t a n t , as o r i g i n a l l y hypothesized. The v a r i a t i o n of f 2 and w i t h the m a t e r i a l - t o - a i r r a t i o was considered under the hypo-theses t h a t f ^ and f ^ remained constant d u r i n g the f l o w process i n each s u b - t e s t and over the range of each t e s t - r u n . V a r i o u s combinations of f ^ and f ^ were considered and the r e s u l t i n g v a l u e s of c a l c u l a t e d s l i p r a t i o s compared w i t h the experimental r e s u l t s . This comparison showed t h a t e i t h e r f , , f^,, or v a r i e d 143 with the m a t e r i a l - t o - a i r r a t i o . At the same time, the c a l c u -l a t e d values of C.p/f2 w e r e considered i n r e l a t i o n to the m a t e r i a l - t o - a i r r a t i o . The v a r i a t i o n of the average s l i p r a t i o and the value of C D / f 2 w i t h the m a t e r i a l - t o - a i r r a t i o strongly suggested that the values of Cp and f 2 both increase with the m a t e r i a l - t o - a i r r a t i o , f 2 from zero and from i t s value under the f r e e - f a l l i n g conditions of a sing l e p a r t i c l e . Maximum or stable values of C D and f 2 are attained at high values of the m a t e r i a l - t o - a i r r a t i o . At these high values of the material-to a i r r a t i o , the two-phase mixture may have become saturated and behaved as a s i n g l e f l u i d . Therefore, i t may be concluded that even though s p e c i f i c values f o r f 2 , f ^ and f ^ were not obtained as o r i g i n a l l y intended, a high degree of r a t i o n a l i t y e x i s t s i n the t h e o r e t i c a l analysis presented i n t h i s t h e s i s ; the r e s u l t s of the experimental work presented herein, together with the c o r r e l a t i o n of the t h e o r e t i c a l analysis with the research of others, warrants considerable further research along the l i n e s of t h i s t h e s i s , with reference to the next section of t h i s chapter. RECOMMENDATIONS FOR FUTURE RESEARCH Research such as that presented i n t h i s t h e s i s normally e n t a i l s the statement of an objective and the formulation of a plan designed to f u l f i l that objective. During the course of the ensuing research, many avenues f o r further or a d d i t i o n a l i n v e s t i g a t i o n become apparent. I f a l l of these avenues were to 144 be followed, the researcher would take on the appearance of Stephen Leacock's character who leapt on h i s horse and rode o f f i n a l l d i r e c t i o n s . I t i s doubtful that anything concrete would be accomplished under such circumstances. Consequently, i t i s appropriate to make recommendations f o r other i n v e s t i -gations, i n the l i g h t of what was revealed during a s p e c i f i c research p r o j e c t . With t h i s i n mind, the following such recommendations are offered: 1. The apparatus which has been assembled i n the Mechanical Engineering Laboratory f o r the purposes of t h i s research should be regarded as being as important and us e f u l i n the f i e l d of p a r t i c l e dynamics and pneumatic conveying as the wind-tunnel i s to aerodynamics. This apparatus should therefore be used as f u l l y as possib l e i n the development of t h i s f i e l d at the U n i v e r s i t y of B r i t i s h Columbia. 2. The i n j e c t o r apparatus should be modified with a view to improving the seals at the ends of i t s enclosure and elim i n a t i n g the e j e c t i o n of material. Such modifica-tions may begin with the use of t e f l o n instead of brass seals and the use of f l e x i b l e t e f l o n scraper-guards i n s i d e the i n j e c t o r enclosure. 3 . The pulse-technique f o r measuring v e l o c i t i e s , as applied 11 by Richardson and McLeman , should be investigated, together with other means, so that particulate phase velocities i n elbows, horizontal runs, and v e r t i c a l runs may be measured. Determination of these velocities w i l l f a c i l i t a t e the calculation of s l i p ratios i n straight and curved ducts having varied geometric orientation. 4 . Work should be conducted to determine the polytropic gas constant as a function of material-to-air ratio and heat-transfer parameters, as they relate to the duct and the duct environment. 5. An automatic or manual control system should be developed whereby the material-to-air ratio may be held constant for a variable a i r flow-rate and, hence, air velocity. This work would entail the close calibration of injector feed rate with r o l l e r speed. A relationship between blower discharge characteristics and injector speed might be devised. On the other hand, perhaps o r i f i c e plate readings could be calibrated against the injector feed-rate characteristics. 6. The duct should be modified such that the geometric con-figuration and diameter can be varied. Removable sections would accomplish this end. Perhaps one-inch and three-inch sections would provide sufficient varia-tion. A single straight and curved section of each dia-meter would be sufficient, provided that they could be 146 located at d i f f e r e n t points i n the present ducto The removable section should also be so designed that i t may be i n c l i n e d at any angle with the h o r i z o n t a l . At the same time, the t r a n s i t i o n from the regular duct to the removable t e s t sections should be located such that conditions s t a b i l i z e before measurements are taken. 7 . A simple apparatus, c o n s i s t i n g of a transparent tube and a strain-gage f i t t e d membrane should be devised whereby the f r e e - f a l l i n g or terminal v e l o c i t i e s of single p a r t i c l e s and groups of p a r t i c l e s might be obtained. This apparatus may be used i n the determination of CJJ f o r si n g l e p a r t i c l e s and f o r i n v e s t i g a t i n g the v a r i a t i o n of Op with the m a t e r i a l - t o - a i r r a t i o . 8 . On the basis of the actual v e l o c i t i e s i n the h o r i z o n t a l , v e r t i c a l and elbow sections of the duct, as determined under recommendations 3 and 5 above, values of f ^ and f ^ should be determined from the r e s u l t i n g s l i p r a t i o s f o r a v a r i e t y of p a r t i c u l a t e materials. I t may be that main-tenance of a constant m a t e r i a l - t o - a i r r a t i o during the v e l o c i t y measurements w i l l maintain constant and f2<> I f the a i r - t o - m a t e r i a l r a t i o i s held at a low value, w i l l have the value f o r f r e e - f a l l i n g p a r t i c l e s . 9 . Having established f ^ and f ^ through recommendation No.8 above, and through No.7, values of fp/Cr> may be com-147 puted on the basis of the experimental data presented i n this thesis or on the basis of new experimental data. Using this information on f2/C D, together with s l i p ratio data for various values of the material-to-air r a t i o , the nature of the function relating C D and f 2 to the material-to-air ratio may be investigated. 10. Once concrete results are obtained for one or two particulate materials, attention should be turned to the effects of particle properties, i n addition to those property effects already taken into account by the free-f a l l i n g or terminal velocity of the particles, insofar as they affect the theoretical analysis presented i n this thesis. SUMMARY This f i n a l chapter has recapitulated the research presented i n the preceding seven chapters of this thesis, drawn a series of conclusions from that research, and recommended steps whereby this research may be pursued, both at the University of B r i t i s h Columbia and elsewhere. What more i s there to say? 148 BIBLIOGRAPHY 1. Zenz, A.A., Othmer, D.F., Fluidization and Fluid Particle Systems, New York, Reinhold Publishing Corp. 1960. 2. Rose, H.E., Barnacle, H.E., Flow of Suspensions of Non-Cohesive Spherical Particles in Pipes, The Engineer, June 14, **" " — — 3. Vogt,.E.G., White, H.R., F r i c t i o n in the Flow of Suspensions, Industrial and Engineering Chemistry, September, 1948. 4. Michell, S.J., Designing a Pneumatic Conveyor, Bri t i s h Chemical Engineering, July, 1957° 5. Hudson, W.G., Conveyors and Related Equipment, New York John Wiley and Sons, Inc., 1954. 6. Engineering Equipment Users Association, Pneumatic Handling of Powdered Materials, London, Constable and Company Limited, 1963. 7. Korn, A.H., How Solids Flow i n Pneumatic Handling Systems, Chemical Engineering, March, 1950. ~~ 8. Farbar, Leonard, Flow Characteristics of Solid-Gas Mixtures, Industrial and Engineering Chemistry, Vol.41, No.6. 9. Zenz, F.A., Minimum Velocity of Catalyst Flow, Petroleum 10. Hitchcock, J.A., Jones, C , The Pneumatic Conveying of Spheres Through Straight Pipes, The B r i t i s h Journal of Applied Physics, June, 1958. 11. Richardson, J.F., McLeman, M., Solids Velocities and Pressure Gradients i n a One-Inch Horizontal Pipe, Trans. Instn Chem. Engrs., Vol .38, 1960. 12. Clark, R.H., Charles, D.E., Richardson, J.F., Newitt, D.M., The Pressure Drop During Horizontal Conveyance, Trans. Instn Chem. Engrs, Vol .30, 1952. ~"~ 13. Soo, S.L., Gas Dynamic Processing Involving Suspended Solids, A.I.Ch.E. Journal, September, 1961. 149 14. Zenz, P.A., Conveyability of Materials of Mixed Particle Size, I&EC Fundamentals, Vol.3, 1964. ~ ~ ' 15. Streeter, Victor L., Fluid Mechanics, New York, McGraw-H i l l , 1958. 16. Glson, Reuben M., Essentials of Engineering Fluid Mechanics, Pennsylvania, International Textbook Company, 1961. 17. Crane Company, Plow of Fluids, Chicago, Technical Paper No.410, 1957^  " ~ 18. Stearns, R.F., Johnson, R.R., Jackson, R.M. and Larsen, C.A., Flow Measurement with Orifice Meters, Van Nostrand, New York, 1951. 19. Kikkawa, S., Utsumi, R., Sakai, K., Nutaba, T., On the Pressure Drop and Clogging Limit i n the  Horizontal Pneumatic Conveyance Pipe, Tokyo, Japan, Bulletin of JSME, November 1965. 20. Murray, J.D., Mathematics of Fluidization, Britain, Journal of Fluid Mechanics 21,465, 1%5. APPENDIX I TABULATED EXPERIMENTAL AND COMPUTED DATA (TABLE' A APPARATUS COMPONENT SPECIFICATIONS COMPONENT NAME Duct Blower Delco Remy Blower Drive G.E. Dodge Injector Injector Drive Collector Chute Receiver Manometer Strain Indicator Baldwin DESCRIPTION" Metal section., 2" std.pipe, 2.06" I.D.,2.375" O.D, Pyrex section, 2.00" I.D., 2 .575" O.D. Elbows, 5° radius,, as jabove; flanged • • Specification unknown; approx. displacement, # cu.ft.per rev; Roots design; 1000 to 2000 rpm. Motor, 10 h.p.1760 rpm, 3ph. 220/440V, variable speed Drive, one fixed 8" dia.pulley, one variable pitch pulley, 5«4" to 9.9" dia. ,ftd.R."type belt; s l i d i n g base to suit. 4~3#" dia.x 12" r o l l s with 40 durometer Neoprene Surface-; 8 -Dodge #" brgs; fabricated.-steel-frame-, brass-seals-. Motor^ .h.p; 1760 rpnt; % hp.Carter Variable speed Driven 0 to 1400 rpm output; 16s 1, 5ihp.H0lr.0yd warm gear; assorted standard" sheaves and chain drives. 16 .G^ A...-G.alv-ani^ ed.iShe:et ,J.with .-plexigla&s. window, sliding ac&ass..port,-.and-sw:inging deflector. 16" Dia. x 60"; 2" in-let, 8" outlet, 6" exhaust, 36" long, #" I.D. -acrylic tubes; hg as H2°^ SR-4 Strain Indicator Potentiometer Leeds-Northrop Temperature Potentiometer, #8692, 152 TABLE B LEGEND OF DUCT DISTANCES MANOMETER "ACTUAL" DISTANCE FROM m A n _ "EQUIVALENT" DISTANCE FROM X - f V t u . n-1 4 Injector n-1 4 Injector 1 8.00 - 8.00 - _ 2 0.20 - = 0.20 -3 2.00 _ 2.00 - -Injector 1.38 - 0.0 1.38 _ 0.0 4 2.54 0.0 2.54 2.54 0.0 2.54 5 8.77 8.77 11.31 8.77 8.77 11.31 6 2.04 10.81 13.35 2.04 10.81 13.35 7 4.67 15.48 18.02 8.13 18.94 21.48 8 2.02 17.50 20.04 2.02 20.96 23.50 19.33 36.83 39.37 19.33 40.29 42.83 10 4.67 41.50 44.04 8.13 48.42 50.96 11 2.00 43.50 46.04 2.00 50.42 52.96 12 79.00 122.50 125.04 79.00 129.42 131.96 13 4.67 127.17 129.71 8.13 137.55 140.09 14 2.00 129.17 131.71 2.00 139.55 142.09 15 6.42 135.59 138.13 6.42 145.97 148.51 16 4.67 140.26 142.80 8.13 154.10 156.64 17 2.00 142.26 144.80 2.00 156.10 158.64 18 79.00 221.26 223.80 79.00 235.10 237.64 19 4.67 225.93 228.47 8.13 243.23 245.77 20 2.00 227.93 230.47 2.00 245.23 247.77 21 9.61 237.54 240.08 9.61 254.84 257.38 Cyclone 2.00 239.54 242.08 2.00 256.84 259.38 Reference; Figure 8 TABLE C No 1 2 3 4 5 6 , Blower P RPM 7 8 9 10 11 12 13 P* Pressure @ Orifice P 0 P, BLOWER CHARACTERISTIC DATA Y A P J P, T , W . Q w 1 x 2 ."-fegg... - " h g a _ P s i s ( P s i g ) ^ "ngg Expacsh Pactor ac ?R #/sec.Cfm #/sec * 1870 14.75 14.90 44.75 1650 11.05 11.20 40.90 1510 9.00 9.15 39.00 1385 7.60 7.75 37.60 1110 4.90 9.95 34.80 1520 9.10 9.30 39.15 7.33 53.9 5.51 30.4 4.50 20.2 3.81 14.5 2.41 5.80 4.47 19.95 3.60 .330 .898 .190 657 2.80 .270 .917 .194 648 2.30 .231 .929 .196 615 1.90 .202 .939 .198 613 1.25 .141 .955 .202 587 2.35 .232 .929 .196 613 .149 121 ,190 154 ,162 131 ,147 119 ,135 109 ,109 88. 1530 1180 1375 1545 1710 1890 2020 V a " Q = 2.90 1.65 2.20 2.90 3.60 4.75 5.40 5.25 35.10 3.20 33.05 4.25 34.10 5.25 35.10 6.35 36.10 7.85 37.70 8.65 58.50 2.58 6.66 3.40 .087 .969 .585 593 .242 196 1.58 2.50 2.20 .050 .982 .592 580 .1815147 2.09 4.37 2.75 .0645.977 .590 583 .212 172 2.58 6.64 3.40 .0826.971 .585 590 .243 197 3.12 9.71 4.00 .1000.965 .582 598 .271 220 4.75 .1260.955 .576 610 .312 253 5.15 .1400.950 .573 618 .332 269 3.86 14.90 4.25 18.10 2.0 2.50 2.04 2.34 3 2.43 1.95 2.50 4.27 2.50 2.50 2.27 2.33 2.20 .187 .157 .148 .135 .113 .149 .248 .1815 .216 .243 .271 .297 .320 .603Y for 1" Orifice .2T1Y for 1.418" Orifice P>, AP' #/sec. n = In In (Pat/ P 1) <Pat ><|5 Px, T at T, 5 Tat R W a W ac = delivery by equation 5(5) T2P at " P a t B S T 11.4P 1 - 219P 1 -292^ at a TABLE D INJECTOR AIR LOSS DATA Nb p1 p4 P 3 P"hw T 3 T 4 R e ^ Wa1 Wa4 "hgg "hga Psig "hgg "hga "hgg 1-3 4-5 °P °P °R centapoclOOO #/sec #sec. Wa1/ Wa4 1840 9.00 "38.86 4.43 8.35 38.21 8.70 4.95 6.00 137 132 592 .0197 140 .0210 .257 .238 .928 1.190 1840 9.15 34.01 4.50 8.40 38.26 8.80 4.90 6.20 153 145 605 .0200 135 .0209 .253 .237 .938 1.240 1840 8.30 38.16 4.09 7.50 37.36 7.80 5.20 5-.60 150 143 603 .0200 125 .0212 .265 .220 .830 1.051 1840 7.00 36.86 3.44 6.40 36.26 6.55 5-75 4.70 147 141 601 .0200 110 .0215 .295 .188 .675 .803 1840 6.35 36.21 3.12 5.70 35.56 5.90 6.10 4.30 143 139 597 .0200 100 .0219 .310 .185 .600 .690 1300 4.40 34.26 2.16 4.10 33.96 4.20 2.60 2.85 108 105 565 .0190 90 .0220 .186 .157 .845 1.085 1300 4.10 33.96 2.01 3.75 33.61 3-90 2.65 2.65 112 108 568 .0190 83 .0221 .193 .149 .775 .990 1300 3.65 33.51 1.79 3.15 33.01 3.25 3.00 2.30 111 107 567 .0190 75 .0222 .205 .135 .660 .755 1300 3.10 32.96 1.52 2.25 32.61 7.85 3.15 1.90 108 105 565 .0190 70 .0223 .220 .125 .570 .596 1500 3.80 33.66 1.87 3.60 33.46 3.60 4.65 2.45 108 105 565 .0190 85 .0221 .265 .146 .550 .523 1500 4.10 33.96 2.02 3.70 33.56 3.80 4.15 2.65 113 109 569 .0195 80 .0221 .255 .148 .580 .630 1500 4.35 34.21 2.14 4.00 33.86 4.10 3.90 2.85 117 113 573 .0196 80 .0221 .250 .153 .626 .721 1500 5.40 35.26 2.65 4.90 34.76 5.10 3.35 3.50 125 121 581 .0197 95 .0220 .222 .172 .775 1.029 1500 5.40 35.26 2.66 5.00 34.86 5.15 3.50 3.50 120 118 578 .0196 100 .0219 .222 .174 .785 1.043 1580 6.10 35.96 3.00 5.65 35.51 5.80 3.80 4.00 126 122 582 .0197 105 .0218 .230 .187 .815 1.040 1690 7.00 36.86 3.45 6.40 36.26 6.40 4.30 4.60 134 129 589 .0198 110 .0215 .245 .202 .825 1.050 P 6 9 0 8.20 58.06 4.02 7.60 57.46 7.90 5.90 5.50 145 158 598 .0200 125 .0212 .220 .2231.000 1.390 Wfl. = Equation 5(5) = a P 4 - 5 P 4 ^1-3 F 3 Wa4 = ^P4-21 P 4 451 fT^ * Seals lubricated and tight, r o l l s not turning, no leakage. w 155 TABLE E EH P W EH w w w H i w PP «=«! EH « oEH H LA I PQ O v LA <j O KM •H •H ft S fH Pi •H PH rA o rA vO oj OJ ON r* L A oo rA o o LA L A OJ IN- rA o • e • • e o • o 0 0 0 O V V rA CO CM OJ y-V ' CM r A vD r A <TVJ VD IA o O O o OJ rA LA O O O rA rA LA P O ON VD O * • • • « e • • • o • ' • VD LA LN O !^  £M rA vD LN o CJN r* LA ^ ' V <r <f LA OJ V V IA LA V CVJ O LA VD o CM 0> rA C0 O LA CVJ V" VD CVJ LA • <• • 0 • • • 0 • e -0 OJ OJ VD IN V rA LA LN LA OJ V" ID CO V- LA OJ o O lA o co IN- LA LA O O po • • • • • • 0 0 • LA VD CO CO CJN LN CVl CVl LN LA 0^ LN LN O OJ rA CN 4- °J CVJ Ov LA CVJ CO 4" OJ <N r- V v- V O IA o LA o O LA O LA o LA CVJ O o O • <• 0 o 0 o 0 « o • rA OJ OJ rA rA rA CVJ CVJ CVl CVJ CVJ . v~ V V- r- r- V V o O LA OJ LA P o CM LN LA o o LA 4- vD fA 0> LA CVJ CO CO CO • • • o • • • 0 • • • . • « VO vD VD LN CO P LA LA VD vD LN CO V V OJ rA LA vD LN V CVJ rA LA VD o VD CVJ p rA speeg t[oq.8A [ CVJ fi it pq CM fi . S I V CVJ •ri Pi II TABLE F1 DATA FROH TEST-HUN SERIES #1 WITH CELLULOSE PELLETS VARIABLE TEST-RUNS 1 2 3 4 5 6 \ rpm 1340 1340 1340 1540 1340 1340 T a t 62 62 62 62 62 62 % T4 Oy 110 112 112 116 119 122 Op 103 104 104.5 105 107 108 T 2 1 oj> 71.2 73.0 76.0 77.5 78.3 79.0 8 9 10 11 12 144.5 1445 1445 1445 1445 1445 64 64 64 64 60 60 117 123 129 134 120 125 107 109 113 116 108 112.1 68.5 73.5 73.5 82.0 74.5 79.0 P a "hga 29.94 29.94 29.94 29.94 29.94 29.94 29.62 29.62 29.62 29.62 29.94 29.94 P 1 "ngg 5.10 5.10 5.40 5.70 6.00 6.45 5.90 6.00 6.30 6.65 6.40 6.90 P-5 4.80 4.80 5.10 5.50 5.90 6.30 5.65 5.80 6.10 6.45 6.20 6.70 P 4 4.70 4.60 4.90 5.20 5.50 5.90 5.25 5.45 5.70 6.20 5.70 6.50 P 9 3.80 3.70 4.00 4.10 4.40 4.70 4.45 4.45 4.70 4.80 4.75 5.00 P 1 2 2.30 2.15 2.20 2.20 2.30 2.50 2.55 2.45 2.50 2.60 2.50 2.60 P 1 8 .50 .40 .40 .45 .50 .50 .50 .50 .50 .50 .50 .50 P 2 1 0.0 0.0 0.0 0.0 0.0 .00 .00 .0 .0 .0 .0 .0 AP 1 - 5 t thw 2.70 2.80 2.60 2.50 2.30 2.20 3.20 3.10 3.0 2.80 3.00 2.85 ^ P 4_5" l l w 2 » 6 5 5.20 5.50 3.80 4.20 4.40 3.15 3.90 4.30 4.60 4.60 4.80 vhmx ^ 0.0 620 575 550 430 170 0 615 580 550 480 350 Wg lb 0.0 1.095 1.645 2.19 3.15 4.11 0 1.37 1.85 2.76 2.40 5.01 t sec. 0.0 82 51 35 22.5 7.5 0 55 40 30 . 27 17 & = 0.0173 #/in^. Sub-test Runs 1 and 7 are for j| /n n M/4„3 free-running conditions. w ^p = 0 ,°526 #/xn^. 0 t< d = 0.160 i n . ^ TABLE F2 TEST-RUN SERIES #2 WITH BARLEY VARIABLE TEST-RUNS 1 2 3 4 5 6 7 8 9 10 11 12 N^ rpm 1690 1690 1690 1690 1690 1690 1420 1420 1420 1420 1420 1420 T a t ° F 67 67 67 67 67 67 67 67 67 67 67 T^ °F 144 154 160 167 169 173 129 134 142 145 146 148 T^ °F 127.5 132.5 137 142 143 146 117.5 121.5 122 123 124 126 T 2 1 °F 72 81 86 91 92 92 75.5 81 82 84 85 86 P Q Mhga 30.31 30.31 30.31 30.31 30.31 30.31 30.31 50.31 30.31 30.31 30.31 30.31 P 1 nhgg 8.20 9.50 10.0 10.60 11.10 11.85 5.90 8.20 8.5 9.00 9.30 9.9 P3 7.80 9.20 9.65 10.2 10.80 11.60 5.60 7.90 8.40 8.75 9.20 9.65 P 4 7.60 8.70 9.4 9.50 10.1 10.65 5.40 7.20 7.7O 8.0 8.40 8.70 P 9 6.90 7.35 7.7 8.0 _ * _ * 4.50 6.10 6.40 6.60 6.8 7.0 P12 3.50 3.80 4.0 4.20 - - 2.50 3.20 3.35 3.40 3.50 3.50 P18 .70 .70 .70 .70 - • .50 .60 .60 .60 .60 .60 P21 0.. 0. 0. 0. - • - .0 .0 .0 .0 .0 .0 AP1_^nhw 4.60 _* * - -' - - 3.60 - - - - -AP^^hw 4.80 _* * - - - - 3.30 - - - - -Vhmx ^ 0 540 420 470 400 315 0 540 460 390 440 530 ws lb 0 1.78 2.32 2.75 3.62 4.25 0 3.56 4.25 5.34 6.02 7.25 t sec. e 0 60 40 35 27 13 0 39 30 21 15 7-5 ep -d -0.0305 #/in 5 0.0486 #/in3 0.121 i n . * readings not taken ** readings not needed Sub-test Runs 1 and 7 are for free-running conditions. H3 ro TABLE P3 TEST-RUN SERIES 0 WITH CELLULOSE PELLETS VARIABLE TEST-RUNS 1 2 3 4 5 6 7 8 9 10 11 12 Nb rpm 1740 1740 1740 1740 1740 1740 1290 1290 1290 1290 1290 1290 op 66 66 66 66 66 66 68 68 68 68 68 68 h op 161 161 164 166 169 172 117 120 123 124 130 130 T 4 Op 156 155 157 159 159.5 161.0 117 118 120 120 122 123 T21 op 88 94 96 96 97.5 98 75 80 81 80 82 83 P a "hga 29*75 29.75 29*75' 29.75 29.75 29.75 29.75 29.75 29.75 29.75 29.75 29.75 P1 "hgg 8.70 9.40 9.8 10.2 10.6 11.1 5.15 6.00 6.40 6.60 6.90 7.20 p 3 8.3 9.1 9.4 9.85 10.2 10.7 4.4.0 5.80 6.30 6.40 6.80 7.00 P 4 8.0 8.60 8.90 9.30 9.60 10.0 4.75 5.50 5.80 6.00 6.30 6.60 P 9 6.75 ,7.1.0 7.40 7.60 7.80 8.30 3.90 4.50 4.70 4.85 5.10 5.15 s P12 3.75 4.00 4.00 4.15 4.30 4.50 2.20 2.50 2.50 2.60 2.60 2.75 P18 0.70 .70 .7 .7 .7 .7 .40 .40 .4 .4 .4 .4 P21 0 0 0 0 0 0 0 0 0 0 0 0 5.70 _ * ' — - . - - 3.10 - - - - - •' c"hw 4.90 - - - - - 2.90 - - - - - • Vhmx ^ 0 575 550 535 510 460 0 480 450 380 380 170 V s lb 0 1.51 1.92 2.27 2.61 3.16 0 2.02 2.95 3.44 3.91 4.46 *e sec. 0 46 36 30 25 17.5 0 35 26 20 1 5 7.5 (2 » 0.0175 #/in^ * Readings not needed -b d = 0.160 i n . Q = 0.0326 #/in^ Sub-test Runs 1 and 7 are for free-running conditions, I ^ VJl KM 00 PRESSURE GRADIENT READINGS. FROM TEST-RUN SERIES. #1 PRESSURE TEST-RUNS ; _ _ _ _ _ _ READING s\ 2 3 4 5 6 7 8 9 10 11 12 P^ "b-SS 5.10 5.10 5.40 5.70 6.00 6.45 5.90 6.00 6.30 6.65 6.40 6.90 P 5 4.80 4.80 5.10 5.50 5.90 6.30 5.65 5.80 6.10 6.45 6.20 6.70 P 4 4.70 4.60 4.90 5.20 5.50 5.90 5.25 5.45 5.70 6.20 5.7O 6.30 p 9 3.80 3.70 4.00 4.10 4.40 4.70 4.45 4.45 4.70 4.80 4.75 5.00 p12 2.30 2.15 2.20 2.20 2.30 2.50 2.55 2.45 2.50 2.60 2.50 2.60 p18 .50 .40 .40 .45 .50 .50 .50 .50 .50 .50 .50 .50 p21 0 0 0 0 0 0 0 0 0 0 0 0 •^^"nw 2.65 3.20 3.50 3.80 4.20 4.40 3.15 3.90 4.30 4.60 4.60 4.80 A P4-6 3.00 3.60 4.20 4.40 4.70 5.00 3.55 4.50 4.90 5.20 5.10 5.60 ^4 - 7 4.70 5.20 5.80 6.20 6.70 6.90 5.50 6.50 6.90 7.3O 7.20 7.50 ^4-8 5.80 4.30 6.90 7.W 8.00 8.50 6.70 7.70 8.20 8.80 8.70 8.90 ^ P4 -9 10.90 12.00 13.40 14.40 16.00 17.70 12.70 14.50 15.70 16.50 16.30 17.70 ^9=10 2,20 2.20 2.50 2.70 2.90 3.10 2.30 2.50 2.50 3.00 3.10 3.20 ^9 -11 1.60 2.20 2.60 3.10 3.60 4.00 1.70 2.30 2.50 3.00 3.70 4.00 ^9-12 22.5 23.2 25.10 26.50 28.30 30.30 26.20 27.50 28.50 30.50 30.50 32.30 A P12 -13 1.30 1.70 1.80 I .90 1.80 1.60 2.00 2700 2.00 2.00 2.10 2.20 ^12-14 1.30 1.70 2.00 2.10 2.30 2.40 1.40 2.00 2.20 2.40 2.40 2.66 ^12^15 3.00 3.5O 4.15 4.50 4.80 5.20 3.40 4.50 5.0 5.50 5.^0 5.70 ^12-16 4.60 5.5O 5.90 6.40 6.70 7.00 5730 6.60 7 .10 7.70 7.50 8.00 ^12 -17 5.40 6.40 7.20 7 . 7 ° 8»20 8.60 6.40 8.00 8.80 9-30 9.10 9.70 ^12-18 23.50 22.20 23.20 24.0 25.30 26.30 27.20 26.50 27.50 28.40 27.80 28.80 ^18 -19 2.10 1.70 1.65 1.50 1.60 1.30 3.00 2.10 2.10 1.70 2.00 1.90 A P18-20 2.40 2.00 2.00 2.00 2.10 2.00 3.20 2.50 2.50 2.50 2.30 2.40 ^18-21 4.50 4.20 4.30 4.50 4.70 5.10 4.70 5.00 5.20 5.30 5.10 5.30 TABLE" H1 AIR PROPERTIES AND PLOW CHARACTERISTICS TEST-RUN SERIES #1 TEST RUN No. W a #/sec Q ^aav. A #-sec. 2 f t ^ X n no R e xlO 5 Rep x104 . f 1 CD 1 .1976 .0827 .0780 .387 .782 . : - 1.674 .924 - .0296 2 .1952 .0822 .0780 .389 .782 1.665 1.674 .905 .378 .0296 .355 5 .1880 .0821 .0780 .391 .782 1.776 1.674 .868 .364 .0298 .355 4 .1799 .0822 .0774 .395 .782 1.610 1.477 .823 .315 .0297 .357 5 .1891 .0833 .0789 .391 .786 1.685 1.477 .878 .384 .0277 .555 6 .1774 .0837 .0788 .394 .794 1.613 1.642 .826 .368 .0272 .355 7 .1865 .0821 .0782 .381 .784 1.642 .875 - .0280 -8 .1865 .0818 .0783 .387 .784 1.645 1.642 .872 .402 .0281 .555 9 .1787 .0825 .0784 .388 .784 1.50371.642 .834 .373 .0281 .355 10 .1711 .0829 .0785 .391 .784 1.510 1.642 .796 .331 .0281 .356 11 .1637 .0833 .0787 .391 .784 1.510 1.642 .759 .346 .0281 .356 12 .1528 .0838 .0789 .392 .784 1.512 1.642 .707 .327 .0283 .356 Note: Test-runs 1 and 7 are for free-running conditions. TABLE H 2 TEST RUN No. V AIR PROPERTIES AND PLOW CHARACTERISTICS TES!T-RUN- SERIES #2 #/sec. #/ff* #/f?3 #~sec i n R R e ep n o x l O 5 x 1 0 4 f t x1 0 ~ 6 1 . 2 2 9 4 . 0 8 6 3 . 0 8 1 0 .393 . 8 1 4 == 1 . 7 6 1 . 0 9 7 ~» . 0 2 8 6 2 . 2 0 4 1 . 0 8 8 0 . 0 8 1 2 . 3 9 8 . 8 1 4 1 . 6 0 1 . 7 6 . 9 6 5 . 2 9 0 . 0 2 8 9 . 3 5 8 3 .1952 . 0 8 8 1 . 0 8 1 0 . 3 9 3 . 8 1 4 1 . 5 7 1 . 7 6 . 9 1 4 .293 . 0 2 8 9 . 3 5 8 4 o 1 8 5 1 . 0 8 8 3 . 0 8 0 7 . 4 0 5 . 8 1 4 1 . 5 5 1 . 7 6 . 8 6 0 . 2 4 4 . 0 2 9 0 . 3 6 3 5 .1772 . 0 8 9 3 . 0 8 1 2 . 4 0 6 . 8 1 4 1 . 5 2 1 . 7 6 . 8 2 2 . 2 6 3 . 0 2 9 0 . 3 6 0 6 c 1 6 6 2 ^ 0 9 0 4 . 0 8 1 7 . 4 0 7 i 8 1 4 1 . 5 2 1 - 7 6 . 7 6 9 . 1 4 4 . 0 2 9 1 . 3 8 8 7 o1929 i 0 8 2 7 . 0 8 0 5 . 3 9 2 . 7 4 9 • - 1 . 8 0 . 8 5 3 •= . 0 3 2 9 -8 . 1 4 0 7 . 0 8 7 3 . 0 8 0 9 . 3 9 4 . 7 4 9 1.47 1 * 8 0 . 6 1 7 .151 . 0 3 4 1 . 3 8 5 9 . 1 5 4 6 . 0 8 7 2 . 0 8 0 8 . 3 9 7 . 7 4 9 1 . 5 4 1 . 8 0 . 5 8 7 . 1 4 9 . 0 3 4 3 . 3 8 6 1 0 > 1 2 4 7 i 0 8 7 5 . 0 8 0 8 . 3 4 8 . 7 4 9 1 . 5 2 1 . 8 0 . 5 4 2 .135 . 0 3 4 5 . 3 9 3 11 i1191 . 0 8 8 4 6 0 8 1 2 . 3 9 9 . 7 4 9 1.49 1 . 8 0 .517 . 1 3 6 . 0 3 4 6 .392 1 2 . 1 0 8 2 . 0 8 9 1 . 0 8 1 5 . 3 9 9 . 7 4 9 1.47 1 . 8 0 . 4 6 9 . 1 0 6 . 0 3 4 9 . 4 1 3 Note; Test-runs 1 and 7 are f o r f r e e - r u n n i n g c o n d i t i o n s . H3 w ro TABLE H3 AIR PROPERTIES AND FLOW CHARACTERISTICS TEST-RUN SERIES #3 TEST RUN No. W a #/sec. #/ff^ $-sec. 2 x1Qb n n o R e x10 5 x10 4 h GD 1 .2318 .0822 .0772 .405 .716 «= 1.91 .946 .0367 — 2 .2187 .0839 .0777 .407 .716 1.67 1.91 .889 .373 .0369 .355 3 .2116 .0841 .0776 .408 .716 1.64 1*91 .858 .361 .0369 .355 4 .2048 .0848 .0780 .409 716 1*63 1.91 6 829 .342 .0369 .356 5 .1982 .0852 .0781 .410 .716 1.62 1.91 .800 .321 .0370 .357 6 .1904 .0858 .0784 .411 .716 1.62 1.91 .767 .276 .0371 .359 7 .1658 .0799 .0770 .391 .760 - 1.98 .744 .0373 -8 .1447 .0815 .0774 .394 .760 1.67 1.98 .646 .236 .0379 .364 9 .1352 .0823 .0770 .395 .760 1.64 1*98 .602 .237 .0380 .363 10 .1305 .0823 .0778 .395 .760 1.67 1.98 .581 .234 .0382 .364 11 .1237 .0824 .0777" .397 .760 1.70 1.98 .548 .173 .0384 .577 12 .1170 .0829 .0779 .3970 .760 1.65 1.98 .518 .203 .0586 .369 Note: Test-runs 1 and 7 are f o r free-running conditions. TABLE_ Jl EXPERIMENTAL SLIP RATIO DATA TEST-RUN SERIES #1 TEST-RUN No. 1 2 3 4 5 6. 7 8 9 10 11 12 FLOW RATES RATE RATIOS m ac #/sec =#/sec W W 0 ,2215 .2872 .5711 .3522 .4078 0 .1498 .2233 .3113 .5785 ,4490 ac W. m v. V , VELOCITIES^ " v l v_ .1527 .1471 .1409 .1483 .1409 .1465 .1463 .1402 .1342 .1284 .1199 m W. ac Ta3 * a3c " aav * amax ' mav " max f t / s e e o f t / s e c . f t / s e c . f t / s e c . f t / s e c . f t / s e c , .689 .512 .379 .422 .345 .977 .628 .431 .539 .267 0 1*450 1*952 2.637 2.368 2.895 0 1*024 1.593 2.319 2.948 3.746 103.4 102.8 99.20 94.80 98.29 91.73 98.37 98.73 93 = 80 89.43 85703 78.92 80.92 80.43 77.62 74.17 77.28 72.85 77.17 77*45 73.58 70.15 66.70 61.91 84.38 81.36 78.29 81.21 76.98 80*70 77.15 73.76 70734 65.47 88.34 85.09 82.41 85.15 81.11 38.80 37.26 39.41 35.22 52 32 83=95 32.68 80.71 32.48 77.36 34.11 73.97 28.84 69 = 03 26.'22 40.62 38.97 41.48 36.92 34.26 33 = 99 33 = 98 35.78 30.53 27.64 AV.SLIP RATIO S a v 0.459 0.458 0.503 0.434 0.422 0.405 0.422 0.463 0.410 0.400 Notes Test-runs 1 and 7 are f o r f r e e - r u n n i n g c o n d i t i o n s . IN El w - A TABLE J2 EXPERIMENTAL SLIP RATIO DATA TEST-RUN SERIES #2 FLOW RATES RATE RATIOS VELOCITIES TEST RUN No. Wm W Q „ m ac #/sec.#/sec w o„ Wm  ac m W_ W m ac v -7 v -7 v v v v a3 ajc aav amax mav max f t / s e c . f t / s e c . f t / s e c . f t / s e c . f t / s e c . f t / s e c . AV.SLIP RATIOS S a v 1 0 .1867 0 115.11 936 71 • - • - - - -2 ,3143 .1662 .5290 1.891 100 .5 8 1 . 8 0 8 7 . 7 6 93^71 42 i 38 45 .25 .483 3 .3667 .1589 A330 2.307 95.97 78.13 84.15 8 0 . 1 6 37.77 4 0 . 4 7 .449 4 .4689 .1507 .3213 3.112 90.82 73.93 79.95 85 .97 4 0 . 9 3 4 4 . 0 0 .512 5 ,5174 . 1 4 4 3 - 2 7 8 9 3 .586 85*95 69 .97 7 6 . 0 2 8 2 . 0 8 34.11 36 .82 .449 6 .8462 .1353.1599 6.253 796 59 6 4 . 8 0 70.70 76.61 47 .79 51.78 . 6 7 6 7 0 .1445• • - 0 ^ 100 i96 7 5 . 6 5 . . _. - - -8 ^4835 .1054.2179 4 . 5 8 9 6 9 . 8 0 52.29 55 .98 59.66 32.51 3 4 . 6 4 .581 9 .-5355 . 1 0 0 8 . 1 8 8 3 5.312 6 6 , 8 1 50.06 5 3 . 5 8 57.12 3 0 . 2 4 32.23 .564 10 .6319 .0935 .1479 6 . 7 6 2 6 1 . 7 0 4 6 . 2 3 4 9 . 6 2 53.02 2 8 . 3 5 30.29 -571 11 .6519 6 0892 .1369 7.307- 58652 4 3 . 6 9 4 7 . 1 1 5 0 . 5 3 25.86 27 .74 .549 12 .7916 .0811 .1024 9 .762 52.59 3 9 . 4 0 4 2 . 6 J " 4 5 . 8 7 " 26.09 2 8 . 0 8 .612 Note: Test-Runs 1 and 7 are f o r free-running conditions h3 ro -p "TABLE J T EXPERIMENTAL SLIP RATIO DATA TEST-RUN SERIES #3 TEST RUN No. FLOW RATES W W Q „ m ac #/sec.#/sec. 'VELOCITIES RATE RATIOS _ _ _ _ _ _ _ _ _ _ _ _ _ W W v , v , v v v v ac m a3 a3c aav amax mav max W W ^  f t / s e c . f t / s e c . f t / s e c . f t / s e c o f t/sec. f t/sec. m ac AV. SLIP RATIOS S a v 1 0_ .1661 - 0 122.2 87.53 ... - - - - -2 .2476 .1567 .6328 1.580 112.9 80.87 86.58 92.29 39.35 41.95 0.455 3 .3026 .1516 .5009 1.996 108.9 78.02 83.73 89.43 37.83 40.41 .452 4 .3533 .1467 .4153 2.407 104.5 74.89 80.60 86.32 37.35 39.99 .463 5 .4041 .1420 .3515 2.845 100.8 72.21 77.90 83.59 37.16 39.87 .477 6 .521 .1365 .2620 3.816 96.09 69 .85 74.50 80.15 39.55 42.54 .531 7 0 .1261 0 89.88 68.34 - - - - -8 .2717 .1100 .4049 2.469 76.83 58.42 61.28 64.13 32.27 33.78 .527 9 .3428 .1028 .2998 3.335 71.16 54.11 56.99 59.88 27.89 29.31 .489 10 .3764 .0993 .2637 3.792 68.64 52.16 54.96 51.72 26.26 27.58 .478 11 .5018 .0941 .1874 5.335 64.99 49.42 52.12 54.82 30.80 32.40 .591 12 .4490 .0890 .1981 5.045 61.15 46.50 49.19 51.88 24.16 25.48 .491 Note: Test-runs 1 and 7 are f o r free-running conditions, TABLE K1 CALCULATED VALUES OF f p AND SLIP RATIOS FOR f , = .200 *4 = .010 TEST RUN SERIES TEST RUN No. f 2 S p SH S V S E S a v c 2 - 1.87 .478 .414 .586 .459 3 - 3.77 .476 .408 .386 .458 4 - 14-.09 .475 .400 .385 .455 1 5 - 5.18 .478 .410 .387 .4$9 6 8.54 .478 .399 .386 .456 8 - 1.4-9 .477 .407 .386 .4$8 9 - 4.07 .475 .399 .386 .456 10 - 98.24 .474 .391 .386 .455 11 5.59 .472 .380 .385 .450 12 2.49 .469 .364 .3843 .445 2 2.00 .469 .408 .578 0.452 3 1.45 .467 .401 .377 0.449 4 1.52 .467 .393 .578 0.448 5 1.04 .465 .383 .377 .445 2 6 1.65 .475 .385 .386 .452 8 .862 .457 .311 .378 .427 9 .776 .454 .295 .377 .421 10 .766 .450 .266 .377 .411 11 .665 .445 .242 .376 .401 12 .696 .443 .203 .379 .386 2 4.70 .478 .417 .386 .460 3 2.60 .477 .412 .386 .459 4 1.68 .476 .406 .386 .457 5 1.40 .475 .401 .386 .456 5 6 1.42 .475 .394 .387 .455 8 1.64 .465 .545 i383 .440 9 1.11 .461 .322 .382 .452 10 1.05 .459 .510 .381 .428 11 1.16 .459 .298 .384 .426 12 .769 .452 .268 .379 .415 TABLE K 2 CALCULATED VALUES OF f P AND S L I P RATIOS FOR f 5 » .250 f4 = . 0 0 9 TABLE K3 CALCULATED VALUES OP f P AND SLIP RATIOS FOR = .300 f4 = .008 TEST TEST D p RUN SERIES RUN No. f2 gp Sv S E s avc 2 - 2.33 .495 " .^33 .342 .468 3 - 6.85 .493 .426 .341 .465 17.14 .489 .418 .341 .462 1 5 - 14.33 .494 .428 .343 .467 6 3.91 .490 .417 .342 .463 8 - 1.75 .493 .425 .342 .465 9 - 7.98 .490 .417 .342 .462 10 8.26 .487 .407 .342 .459 11 3.09 .484 .396 .341 .455 12 1.84 .479 .378 .341 .449 2 1.51 .487 .427 .333 .460 3 1.15 .484 .419 .333 .457 4 . 1.08 .483 .411 .334 .455 5 .876 .479 .400 .333 .451 2 6 1.298 .486 .399 .342 .457 8 .775 .460 .322 .336 .426 9 .712 .455 .304 .335 .418 10 .716 .447 .274 .336 .406 11 .636 .439 .249 .335 .394 12 .684 .431 .207 .338 .376 2 2.77 .495 .437 .341 .469 | 3 1.85 .494 .431 .341 .467 4 1.31 .442 .425 .341 .46$ 5 1.13 .491 .419 .342 .463 -3 6 1.15 .489 .411 .342 .461 8 1.33 .473 .359 .340 .441 9 .969 .466 .334 .339 .4,32 10 .930 .462 .320 .339 .427 11 1.031 .461 .307 .342 .424 12 .719 .449 .275 .338 .408 TABLE K4 CALCULATED VALUES OF f P AND SLIP RATIOS FOR = 0 . 2 0 f 4 = 0.00275 TEST TEST RUN SERIES RUN NO. f 2S p SV S E 2 - 6.71 .620 .511 .390 .575 5 5.09 .618 .501 .389 .572 4 1.52 .615 .489 .301 .568 5 2.21 .619 .505 .391 .575 1 6 1.01 .615 .487 .391 .568 8 - 2.51 .618 .499 .390 .571 9 2.88 .615 .487 .390 .568 10 1.35 .612 .474 .390 .564 11 0.928 .609 .458 .390 .559 12 .703 .605 .455 .390 .551 2 o590 .613 .507 .388 .568 5 .492 .610 .496 .381 .564 4 .472 .608 .483 .382 .561 5 .406 .605 .448 .382 .556 2 6 .547 .610 .461 .391 .561 8 .386 .584 .359 .387 .523 9 .587 .578 .337 .386 .514 10 .373 .569 .299 .387 .498 11 .343 .561 .269 .387 .485 12 .379 .551 .220 .392 .458 2 0.841 .621 .516 .389 .576 5 0.673 .619 .508 .389 .574 4 .541 .618 .499 .390 .571 5 .488 .615 .489 .390 .569 5 6 .497 .614 .478 .391 .566 8 .575 .596 .406 .390 .542 9 .459 .589 .574 .389 .550 10 .448 .585 .557 .389 .523 11 .495 .585 .559 .395 .518 12 .375 .571 .301 .390 .499 TABLE K5 CALCULATED VALUES OF f ~ AND SLIP RATIOS FOR f 5 = .250 f4 = .00300 TEST TEST V o RUN SERIES RUN No. f 2 S p S E Savc 2 - 6.234 .605 .505 .364 .558 3 3-630 .602 .496 .365 .554 4 1.708 .598 .485 .363 = 551 2.53 .605- .498 .565 .556 1 6 1.12 .599 .485 .365 .551 8 - 2.567 .602 .494 .564 .555 9 3.56 .598 .482 .564 .551 10 1.49 .595 .469 .564 .546 11 1.05 .590 .454 .564 .541 12 .779 .583 .429 .564 .552 2 .647 .598 .502 .556 .551 3 .558 .594 .491 .355 .547 4 .518 .592 .479 .556 .544 5 .445 .587 .464 .355 .409 2 6 .602 .592 .457 .565 .545 8 .427 .561 .557 .361 .505 9 .405 .554 .535 .560 .495 10 .415 .544 .297 .361 .477 11 .383 .534 .268 .381 .463 12 .425 .522 .219 .365 .438 2 .928 .606 .511 .364 .559 .741 .605 .505 .565 .557 4 .594 .601 .494 • 364^  .554 5 .555 .599 .485 .564 .551 3 6 .546 .597 .474 .365 .549 8 .635 .576 .405 .564 .525 9 .509 .567 .572 .564 .511 10 .498 .561 .554 • 565 .505 11 .549 .559 .537 = 567 .498 12 .417 .545 .299 .564 .479 TABLE K6 CALCULATED VALUES OP f~ AND SLIP RATIOS FOR f , * , n f4 - .00325 TEST TEST CD Ap RUN RUN SERIES No. f 2 S p SH S V S E S a v c 2 - 5.880 .589 .500 .343 .542 5 4.250 .587 .491 .345 .559 4 1.907 .585 .481 .342 .555 5 2.88 .588 .493 .544 .541 1 6 1.24 .585 .478 .344 .555 8 = 2.617 .586 .490 .543 .559 9 5.890 .582 .428 .345 .555 10 1.660 .578 .465 .344 .531 11 1.157 .574 .451 .545 .525 12 .856 .566 .426 .345 .516 2 .7046 .585 .496 .555 .556 5 .585 .579 .486 .334 .551 .565 .576 .474 .555 .528 .483 .571 .459 .355 .523 2 6 .657 .575 .455 .544 .526 8 .469 .559 .555 .559 .485 9 .443 .552 .334 .559 .475 10 .459 .529 .296 .540 .459 11 .424 .509 .267 .539 H h h 12 .473 .494 .219 .344 .419 2 1.018 .591 .506 .343 .544 3 .809 .589 .498 .345 .541 4 .647 .586 .489 .345 .558 5 .585 .583 .481 .545 .556 5 6 .595 .581 .470 .344 .553 8 .697 .557 .401 .545 .506 9 .559 .546 .569 .345 .493 10 .548 .541 .552 .545 .485 11 .606 .557 .556 .346 .480 12 .462 .521 .298 .542 .461 TABLE K7 CALCULATED VALUES OF f 2 AND SLIP RATIOS F 0 R f 3 = .300 172 RUN f, .005 TEST RUN No. _ 2 f 2 S p S E S a v c 2 - 3 .374 .546 .471 .343 .509 3 19.710 .544 .463 .342 .506 4 3 .346 .540 .453 .342 .502 5 6 . 9 0 4 .545 .465 .344 .507 6 1.851 .541 .452 .343 .503 8 - 2.107 .543 .462 .243 .506 9 13.880 .540 .451 .343 .502 10 2 .704 .537 .440 .343 .498 11 1 .634 .533 .427 .343 .494 12 1.151 .526 .406 .343 .486 2 .949 .538 .466 .334 .501 3 .767 .535 .456 .334 .498 4 .732 .533 .446 .335 .495 • 5 .617 .529 .434 .335 .490 6 .861 .534 .429 .343 .495 8 .577 .504 .341 .338 .459 9 .538 .497 .321 .338 .450 10 .551 .487 .287 .338 .436 11 .501 .478 .259 .338 .422 12 .550 .466 .214 .342 .401 2 1 .469 .547 . 4 7 5 .342 .50$ 3 1.111 .545 .468 .342 .507 4 .858 .543 .461 .342 .505 5 .761 .541 .453 .343 .502 6 .776 .539 .444 .344 .500 8 .902 .519 .383 .342 .477 9 .699 .510 .355 .341 .466 10 .679 .505 .339 .341 .459 11 .750 .503 .324 .344 . 4 5 5 12 .553 .489 .289 .341 .438 TABLE K8 CALCULATED VALUES OF f P AND SLIP RATIOS FOR f 5 F 4 » .001 TEST TEST RUN SERIES RUN No. S H Sv SE Savc 2 8.498 .655 .557 .299 .579 3 1.730 .648 .545 .299 .574 4 1.117 .641 .531 .299 .567 5 1.419 .650 .547 .308 .575 1 6 .826 .639 .527 .300 .566 8 - 7.614 .647 .543 .299 .573 9 1.690 .639 .527 .299 .566 10 1.038 .630 .511 .300 .558 11 .799 .619 .492 .299 .549 12 .650 .602 .461 .299 .534 2 .509 .652 .556 .292 .574 3 .439 .643 .541 .292 .568 4 .430 .635 .525 .292 .561 5 .381 .525 .506 .292 .552 2 6 .508 .622 ^495 .301 .551 8 .419 .553 1377 .296 .488 9 .407 .538 .353 .296 .474 10 .439 .513 .310 .296 .450 11 .423 .494 .277 .296 .429 12 .500 .463 .226 .299 .396 2 .687 .659 .565 .299 .583 3 .576 .653 .553 .299 .578 4 .482 .647 .542 .299 .573 5 .445 .641 .531 .299 .567 3 6 .459 .633 .516 .301 .561 8 .567 .584 .430 .299 .518 9 .484 .563 .394 .299 .498 10 .485 .551 .374 .299 .487 11 .545 .540 .353 .302 .477 12 .443 .515 .312 .299 .452 TABLE K9 CALCULATED VALUES OP £ 0 AND SLIP RATIOS F 0 R f3 - 6 7 5 f4 - .0002 TEST RUN 3ERIES TEST RUN No. V o f2 Sp % « 7 S E s avc 2 - 4.500 .6546 .587 .259 .566 3 1.568 .645 .575 .258 .578 4 1.092 .629 .556 .258 .548 5 1.326 .645 .574 .259 • 559 1 6 .837 .627 .552 .259 ' .546 8 - 95.12 .641 .570 .258 .556 9 1.576 .626 • 552 .259 .546 10 1.045 .611 .555 .259 .534 11 .859 .594 .512 .258 .522 12 .7188 .567 .478 .258 .501 2 .516 .655 .587 .252 .563 3 .453 .641 .570 .251 .553 4 .450 .626 .551 .252 .542 5 .408 .608 .529 .252 .529 2 6 .547 .597 .515 .259 .524 8 .509 .494 .587 .254 .441 9 .509 .472 .561 .255 .423 10 .575 .456 .516 .255 .395 11 .574 .408 .282 .251 .368 12 .722 .564 .228 .252 .529 2 .685 .662 .596 .258 .571 5 .585, .651 .583 .258 .564 4 .500 .675 .640 .569 .258 5 .468 .629 .556 .259 .548 3 6 .488 .616 .559 .259 .558 8 .648 .559 .443 .257 .479 9 .579 .507 .404 .257 .453 10 .592 .490 .383 .256 .459 11 .681 .475 .362 .259 .425 12 .581 .457 .318 .255 .594 CALCULATED VALUES FOR TABLE K 1 Q OF f 2 AND SLIP RATIOS f 3 - o50 f 4 - .0005 TEST TEST V p RUN SERIES RUN No. f 2 S p SV S E ^avc 2 5.037 .675 *575 .289 .589 3 1 .497 .666 .561 .288 .585 4 1.011 o656 .546 .288 .575 5 1 .25 .667 .563 .289 .584 1 6 .764 .654 .542 .289 .574 8 - 1 6 . 7 8 .665 .559 .289 .582 9 1 .48 .654 ,542 .289 .573 10 .953 .643 .524 .289 .564 11 .749 .629 .505 .289 .553 12 .626 .608 .472 .289 .536 2 .476 .673 .574 .281 .586 3 .414 .663 .559 .281 .578 4 .407 .652 .541 .282 .569 5 .365 .639 .520 .282 .558 2 6 .487 .652 .507 .289 .555 8. .417 .550 .583 .285 . 4 8 5 9 .409 .553 .557 .285 .469 10 .448 .504 .314 .285 .443 11 .457 ,481 .280 .284 .421 12 .526 .446 .227 .287 .385 2 .654 .680 .583 .288 .594 3 .558 .673 .571 .288 .587 4 .455 .664 .558 .288 .581 5 .422 .656 .546 .289 .575 3 6 .456 .647 .529 .289 .567 8 .551 .587 .458 .288 .517 9 .479 .562, .400 .288 .495 10 .483 .548 .579 .287 .484 11 .547 .554 .559 .291 .472 12 .452 .505 .516 .287 /i /i i\ O T IT TABLE M1 RATIOS OF POLYTROPIC CONSTANTS AND PLOW RATES TEST-RUN SERIES #1 TEST "RUN No. POLYTROPIC CONSTANTS PLOW RATES R1 R2 R? R4 R6 Rr^  R8 1 1.000 0 1.000 1.000 1.000 2 .9944 • .0056 .6854 .4057 .9937 2.435 .6853 1.679 .6811 3 1.061 - .0385 .4914 ..3249 .9918 2.928 .5080 1.499 .4873 4 .9615 + .6385 .3476 .2520 .9929 3.611 .3766 1.369 .3451 1.140 - .1404 .3958 .2783 .9901 3.335 .4181 1.408 .3919 6 1.092 - .0918 .3022 .2246 .9952 3.876 .3438 1.339 .3007 7 1.000 0 1.000 1.000 1.000 8 1.002 - .0018 .9802 .4959 .9965 2.017 • 9733 1.969 .9768 9 0.9158 .0842 .5986 .3677 1.005 2.606 .6309 1.636 .6016 10 .9195 .0875 .3920 .2739 1.009 3.350 .4353 1.445 .3957 11 .9194 .0806 .2931 .2189 1.015 4.009 .3444 1.359 .2976 12 .9207 .0793 .2142 .1690 1.021 4.847 .2727 1.294 .2188 R1 - m R ? = R 5 0 _ac Wra + W D „ m ac m R 2 - (n Q - n)/n 0 5 ac 0 Va3co R 8 = R 6 o Wac W wacc va3c Wm m R3 " Wac 0 va5c Ra= W Q „ 6 ac + W m m 0 a5co R 9 = W ac e V a c Wm va3co W aco va3c W aco m TABLE M2 RATIOS OF POLTTROPIC CONSTANTS AND FLOW RATES TEST-RUN SERIES #2 TEST RUN No. P O L T T R O P I C CONSTANTS FLOW RATES R-1 1.000 0- 1.000 1.000 1.000 2 .9063 .0957 .4615 .3019 1.019 2.947 .5590 1.558 .4705 3 .8885 .1115 .3615 .2521 1.021 5.576 .4240 1.465 .5688 4 .8765 .1235 .2556 .1919 1.023 4.206 .3288 1.352 .2594 5 .8602 .1598 .2082 .1628 1.035 4.745 .2885 1.323 .2155 6 .8599 .1401 .1106 .0955 1.048 7.601 .1676 1.216 .1159 7 1.00 0 1.000 1.000 1.000 8 .8146 .1854 .1507 .1237 1.055 5.895 .2299 1.285 .1589 9 .8537 .1463 .1246 .1048 1.054 6.654 .1985 1.255 .1513 10 .844-7 .1553 .0904 .0787 1.058 8.215 .1565 1.215 .0956 11 .8255 .1747 .0790 .0695 1.069 8.879 .1465 1.215 .0845 12 .8157 .1843 .0554 .0484 1.077 11.59 .1104 1.188 .0575 R i -R 2 = (n Q - n)/nQ R 3 y v x„ 3 ac o ape V v , m a3co R,. = Rx Wm 4 3 o _2L Ro = m ac R 5 = JJac 0 va5co W v -, aco a3c R 5 " Wac ? Vm „va5co W aco a3c f R 5 o ! a c R o = R A W Q„ 8 6 0 ac m R 9 = ^ac o ^ac Waco Wm H3 W ro •v3 TABLE M3 RATIOS OF POLYTROPIC CONSTANTS AND FLOW RATES TEST-RUN SERIES #3 TEST RUN No. POLYTROPIC CONSTANTS FLOW RATES R1 R 2 R4 R5 R6 E7 R8 R 9 1 1.000 0.0 1.000 1.000 1.000 2 .8708 .1292 .5847 .3581 1.021 2.634 .6461 1.667 .5970 3 .8614 .1386 .4465 .2975 1.024 3.068 .5129 1.537 .4572 4 .8553 .1447 .3553 .2511 1.032 3.518 .4287 1.461 .3668 5 .8493 .1507 .2899 .2145 1.036 .3986 .3643 1.401 .3005 6 .8455 .1545 .2061 .1633 1.044 5.030 .2737 1.318 .2153 7 1.000 0.0 1.000 1.000 1.000 8 .8422 .1578 .3462 .2464 1.021 3.541 .4133 1.434 .3533 9 .8258 .1742 .2374 .1826 1.029 4.464 .3087 1.338 .2444 10 .8434 .1857 .2014 .1594 1.031 4.940 .2718 1.303 .2076 11 .8580 .1420 .1355 .1142 1.0316 6.535 .1934 1.225 .1398 12 .8306 .1694 .1349 .1125 1.037 6.27I .2056 1.243 .1399 R1 = n / no R 4 R 3 0 21 y m m ac R ? = R 5 0 _ c Wm R 2 = (n 0 - » y no R^ = W W v ac o V aco a3co a3c R8= R6 0 Wac W m R 3 - Wac 0 va3c R 6 = W ac m R 9 - W Q„ W o„ ac 0 ac m va3co W aco W W aco m 179 TABLE IT PARTICLE DRAG COEFFICIENT AND REYNOLDS NUMBER RATIOS TEST TEST Y—— RUN SERIES RUN No. V R e p CD Rep p R e p 2 J V % x 10" 4 x 10 4 ' x 10 5 Y x 1 0 2 2 .94-02 .1545 .1717 .2198 3 .9774 .1293 .1674 .2170 4 1.1308 .1126 .1525 .2067 5 .924-6 .1365 .1757 .2211 1 6 .9644 .1309 .1689 .2179 8 .8847 .1427 .1789 .2244 9 .9516 .1327 .1704 .2189 10 1.0778 .11772 .1572 .2100 11 1.0287 .1230 .1619 .2154 12 1.0893 .1165 .1561 .2095 2 1.2365 .1038 .1445 .2007 3 1.222 .1048 .1455 .2014 4 1.4876 .0884 .1291 .1887 5 1.5721 .0947 .1554 .1958 2. 6 2.6900 .0559 .0951 .1549 8 2.5560 .0581 .0956 .1575 9 2.5940 .0574 .0949 .1568 10 2.9030 .0551 .0895 .1509 11 2.8960 .0531 .0896 .1511 12 3.9080 .0456 .0772 .1567 2 .9523 .1326 .1705 .2189 " ~ | 3 .9835 .1285 .1667 .2166 4 1.0415 .1216 .1607 .2125 5 1.1099 .1145 .1545 .2080 3 6 1.3017 .0991 .1598 .1975 8 1.559 .0859 .1265 .1866 9 1.552 .0862 .1269 .1869 10 1.555 .0855 .1259 .186 11 2.1810 .0652 .1040 .1661 12 1.817 .0751 .1151 .1765 APPENDIX I I COMPUTATIONS AND COMPUTER PROGRAM 181 TABLE P RELATION OF PROGAM VARIABLES TO THOSE IN TEXT REFERENCE 2 LIST OP VARIABLE SYMBOLS PROGRAM TEXT COMMENTS or DESCRIPTION VARIABLE STATEMENT No. VARIABLE FORMULA No. D 1 D _ Duct diameter R 2 R - Gas constant G 3 S - Gravity F 3 F4 4 f 3 f4 - F r i c t i o n factor 5 - F r i c t i o n factor MM 6 - Counter DM1 7 Pb - Bulk density DM2 7 ?P - Solid density ASP 7 A /S^ P P d - Area ratio DD 7 - Particle diameter N 11 - - No. of Sub-tests P10 - P210 13 P10 ~ P210 - Free-running pressure P1 - P21 13 P1 " P21 — Two-phase pressure PA 13 P a t - Atmospheric pressure P13 13 ^1=3 - Pressure drop,taps 1-3 P14 13 A P 4 - 5 - Pressure drop,taps 4-5 T40 - T210 13 T40 " T210 Free-running temperature T4 - T21 13 T4 " T21 - Two-phase temperature TA 13 T x a t - Atmospheric temperature XMS 13 W s - Material weight i n line T 13 - Time to empty hopper VHMAX 13 v hmx - Volume in hopper BRPM 13 Nb <= Blower speed, rpm TABLE P (contd). 182 PROGRAM TEXT VARIABLE STATEMENT No. VARIABLE FORMULA No. COMMENTS or DESCRIPTION DPO, DP 4-6,4? 6(8) Free-running pressure drop TAVO, TAV 50,51 T T avm' av 6(7b) Average temp.drop VISCO, vise 52,55 Pa3o' ^ a5 6(7a) Air viscosity RHOO, RHO 54,55 6(3) Air density WAO, VA 56,57 Uao» V a 5(4) Air flow rate VA30, VA3 60,61 v a5o' v a3 =, Air velocity GAMMA 65 5(8) Injector efficiency VA3CO, VA3C 66,67 va3co» va3c 6 ( 5 ) Corrected air velocity REO, RE 70,71 Reo> R e 6(9) Duct Reynolds No. FIO, F1 72,100 f1Q> f1 3(45) Darcy-Weisbach f r i c t i o n XNO, XN 73,74 6(5) Polytropic gas constant DPR 75 = - Convenience factor CC 76 «= Correction factor for f^ WM 101 Wm 5(15) Mass flow rate AMR 102 a m Air to material ratio VAAV 103 V aav 6(11) Average a i r velocity VMAV 104 v may 6(12) Average material velocity S 105 S a v 6(14) Average s l i p factor, exp. VAMX 106 v amx - Maximum air velocity VMMX 107 V TnTYl-V Maximum material v e l . RHOAV 110 min A-£aav 6(15) Average air density REP 111 Rep 6(10) Particle Reynolds No. CD 112 CD 5(39) Particle Drag Coeff. 183 TABLE P (Oontd). PROGRAM TEXT COMMENTS or DESCRIPTION VARIABLE STATEMENT No. VARIABLE FORMULA No. B1 ,B2,B3,BA- 113-116 B1 ,B2 3(34) Convenience factors SH 117 SH» Ss 3(34) Horizontal s l i p ratio SV 120 V S s 3(34) Vertical s l i p ratio E1 . - E7 121-127 3(37) Convenience factors * SE 131 S E ' S c 3(37) Elbow s l i p ratio SE1 130 S E ' S c 3(37) Estimate of S„ c S1,S2,S3 132-134 - 3(37) Convenience factors PF, FF1 135-136 - 3(37) Convenience, SAV 143 S avc 3(38) Average s l i p ratio, calculated VL 144 3(50), 6(2) Summation of ve r t i c a l length factors EL 145 3(50), 6(2) Summation of ver t i c a l length factors HL 146 3(50), 6(2) Summation of horizon^ al length factors OF1,CF2,CF3 147-151 - 6(2) Convenience factors A(I,1) 153 a 6(2) Simultaneous equation coefficients A(I,2) 154 b 6(2) Simultaneous equation coefficients A(I ,3 ) 155 c 6(2) Simultaneous equation coefficients A(I,4) 156 d 6(2) Simultaneous equation coefficients RI-RZ4 157-206 - - Various parameters 'X(IJ,3) 272 6(2) ( C D A p ) / ( f 2 S p ) X(IJ,II) 301 X 2 ' X 3 6(2) f ? , (x 1+1)f 4 XSUM(I) 316 — Summations o f L , X 2i ' Z 3 184-TABLE P (Contd) XMEAN(I) XSTD(I) STDD(I) PROGRAM TEXT VARIABLE STATEMENT VARIABLE FORMULA No. No o 317 321 322 COMMENTS or DESCRIPTION Arithmetic mean of » ^2' ^3 Convenience f a c t o r of 5 C y j j ' Standard deviation of X^j, X 2, X, * SE i s solved by an i t e r a t i v e procedure. ** Statements 250 to 311 solve a l l of the combinations of three equations i n three unknowns r e s u l t i n g from 10 sets of data, i . e . 120 combinations. The Arithmetic mean and standard deviation f o r each of the unknowns i n equation 6(2), as derived from the 120 s o l u t i o n s , i s then computed. 185 FORTRAN PROGRAM SJOB 59165 N.A. JOHNSON STIME 19 ^FORTRAN C This Program i s designed to calculate the constants in the Model C A(X^) - B(xp) - C(x,) = D and then solve each combination ^ of 5 equations for x^,x 2 and x^ C If ASP - 1. Then i t i s unknown and GD/F2 = (CD*ASP)/F2 1 D = 0.1715 2 R = 55.5 3 G - 32.2 C D=FEET, R=FEET/DEG.RANK., G=FEET/SEC.**2,BRPM=BLOWER RPM, DM1=BULK C DENSITY IN LB/IN.**3, P10=FREE-RUNNING PRESSURE IN INHGGGE, AT PT.1, P1=CONVEYING PRESSURE AT PT.1,T10=FSEE-RUNNING TEMP. AT PT.1, ETC. C VHMAX=HOPPER VOL. AT STEADY STATE, IN.**3,T=N0.SECS TO EMPTY VHMAX, ASP=MEAN RATIO OF PARTICLE X-SECTION AREA/ SURFACE AREA. C PA IN INHGABS. 4 F5 = 0.0250 5 F4 = 0„275E=02 6 - 200 MM s 1 7 20 READ (5,1) DM1, ASP, DD, DM2 10 1 FORMAT (4F10.5) C DD=PARTICLE DIA. IN FEET, DM2=S0LID DENSITY IN LBS/FT**5 11 N = 10 C 12 DO 15 1 - 1, N 15 2 READ (5,5)P10,P30,P40,P50,P210,P1,P5,P21,PA,P15,P45, 1 T40,T210.T4,T21,2A,XMS,T, VHMAX, BRPM,T50,T5 14 3 FORMAT (11F6,2. ,/8F8.2,F8.1,/2F8.2) 15 Z « 0.0 16 72 FORMAT(20X,15H BARLEY TESTS,///) 17 87 WRITE (6,4) 20 4 FORMAT (1H1,5X,4HBRPM, 7X,5HDMI,6X,5HP10,6X,5HP30,6X, 1 3HP40,6X,3HP50,5X,4HP210,7X,2HP1\7X,2HP5S6X,3HP21, 2 7X,2HPA,6X,3HP13,6X,3HP45,6X,5HASP,//) 21 WRITE (6 ,5) BRPM,DM1, P10,P50,P40,P50,P210,P1,P5,P21,PA, 1 P13,P45,ASP 22 5 FORMAT (2X,F10.4,F10.5,11F9.5,F10.5//) 23 WRITE (6,6) 24 6 FORMAT (5X,3HT30,4X,4HT210,5X,2HT3?5X,3HT21,6X,2HTA,6X, 1 3HXMS,6X,1HT,4X,5HVHMAX,5X,3HT40,5X,2HT4//) 25 WRITE (6,7)T50,T210,T5,T21,TA,XMS,T,VHMAX,T40,T4 26 7 FORMAT (2X,10F8.2,///) 27 P30=P30 +PA 30 P40=P40 +PA 31 P50=P50 +PA 32 P210=P210 + PA 53 P3 = P3 + PA 186; P21 = P21 + PA T30 = TJO + 460 T40 = T40 + 4 6 0 T210 = T210 + 4 6 0 T3 = T3 + 460 T4 = T4 + 4 6 0 T21 = T21 + 4 6 0 TA = TA + 460 T3 = T 3 - T30 + T40 T30 = T40 T3 I S CORRECTED TO SAME DROP AS T 3 0 - T 4 0 DPO - P 3 0 - P 2 1 0 DP = P3 - P21 TAVO = ( T 3 0 + T 2 1 0 ) / 2 TAV=(T3+T21)/2 V ISCO -0 .0001262*(1 ./ ( „555*TAVO+120. ) )* (TAVO/460. )* *1.50 VISC = . 0 0 0 1 2 6 2 *(1 ./ ( .555*TAV+120. ) )* (TAV/460. )* *1.50 VISC=MEAN CONVEYING AIR V ISCOSITY I N ( L B - S E C ) / F T * * 2 RHOO=70.9*P30/(R*T30) RHO= 70.9*P3/(R*T3) RHO * AIR DENSITY AT P T . 3 , LB/FT**3 WAO=(PA/(175.*TA))*(BRPM + 2 «76*(P10)**2) -108.*P10 -292. ) ¥ A = ( P A / ( 1 7 5 . * T A ) ) * ( B R P M + 2 . 7 6 * ( ( P 1 ) * * 2 ) - 1 0 8 . * P 1 - 2 9 2 . ) WA » AIR DEL IVERY, L B / S E C . VA30=43.3*WA0/RH00 VA3= 43.3*WA/RH0 VA3 - A IR VELOCITY PRIOR TO FEEDER, FT/SEC<> P10=P10 + PA P1 = P1 +PA GAMMA- 0 . 8 0 * ( ( ( P 4 5 ) * ( P 4 0 ) ) / ( ( P 1 3 ) ) * ( P 1 0 ) ) ) * * 0 . 6 5 GAMMA . FEEDER EFF IC IENCY VA3C0 = GAMMA*VA30 VA3C = GAMMA*VA3 VA3C = A IR VELOCITY CORRECTED FOR FEEDER LOSSES REO =(.231*WAO*GAMMA)/VISCO RE«s( .231 *WA*GAMMA)/VISC RE= REYNOLDS NO, (CONSTANT FOR PROCESS) F 1 0 - 0 . 0 1 0 8 4 * ( R E 0 * * 0 . 0 3 0 6 7 ) * E X P ( 2 1 . 8 / ( R E 0 * * 0.3586)) X N 0 = ( A L 0 G ( P 3 0 / P 2 1 0 ) ) / A L 0 G ( ( T 2 1 0 * P 3 0 ) / ( T 3 0 * P 2 1 0 ) ) XN =(ALOG (P3/P21) )/ALOG(0?21*P3)/(T3*P21) ) XN = POLYTROPIC GAS CONSTANT DPR = DP/(2.*XN*P3) CC = DPO/(„336*F10*(VA3CO**2)*RHOO*(1.+DPR)) F 1 0 = CC*F10 F 1 = 0 , 0 1 0 8 4 * ( R E * * 0 , 0 3 0 6 7 ) * E X P ( 2 1 o 8 / ( R E * * 0 o 3 5 8 6 ) ) * C C F1 = DARCY-WEISBACH FLOW FRICTION FACTOR WM = (1.145*VHMAX*DM1)/T WM = FEED RATE , L B / S E C . AMR = WA/WM AMR = AIR/MATERIAL RATIO VAAV » VA3C*(1o+DPR) VMAV - (240.*WM)/XMS 1 8 ? 105 S = V M A V / V A A V 1 0 6 V A M X » V A 3 C * ( 1 . + 1 . * D P E ) 107 V M M X = S * V A M X C S = S L I P R A T I O = M A T E R I A L V E L O C I T Y / A I R V E L O C I T Y 1 1 0 R H O A V = ( 3 5 • 4-5/R) * ( P 3 / T 3 + P 2 1 / T 2 1 ) 1 1 1 R E P = ( ( V A A V - V M A V ) * D D * R H 0 A V ) / ( V I S C * G * 1 2 . ) 1 1 2 C D = ( 7 . 0 0 * E X P ( 1 . 1 5 0 * R E P * * 0 . 1 8 7 5 ) ) / R E P * * 1 . 0 1 6 115 B 1 = ( D D * D M 2 ) / ( C D * R H 0 A V * D ) 1 1 4 5 4 B 2 » 1 . - 9 6 . * F 4 . B 1 115 33 B 5 = 1 . - 1 9 2 . * G * F 3 * B 1 D / V A A V * * 2 1 1 6 B 4 » 1 . - 1 9 2 . * G * B 1 * D / V A A V * * 2 117 S H = ( 1 . / B 2 ) * ( 1 . - S Q R T ( 1 , - B 2 * B 3 ) > 1 2 0 S V = ( 1 , / B 2 ) * ( 1 . - S Q R T ( ' I . - B 2 * B 4 ) ) 1 2 1 E 1 = ( 1 9 2 . * * D D . D M 2 * G * F 3 ) / ( C D * R H 0 A V * V A A V * * 2 ) 1 2 2 E 2 = ( V A A V * * 2 / ( 3 . * G ) ) * * 2 123 E3=.50 1 2 4 E 4 = 0 . 0 125 E 5 = . 4 2 5 * V A A V * * 2 1 2 6 E 6 = . 6 3 7 * E 1 / J ? 3 127 E 7 = F 4 * V A A V * * 2 * E 1 / ( 2 . * G * D * F 3 ) 130 S E 1 - . 3 0 131 32 S E = S E 1 1 3 2 S 1 ~ E 2 * S E * * 4 . - E 5 * S E * * 2 * E 4 + E3 133 S 2 = E 1 * S 1 * * . 5 + E 6 * E 7 * S E * * 2 134 S 3 = 4 . * E 2 * S E * * 3 . - 2 c * E 5 * E 4 * S E 135 F F - S E - 1 . + S 2 * * . 5 1 5 6 F F 1 «1 . -t- ( . 2 5 / S 2 * * . 5) * ( 4 . *E7* S E + E 1 * S 3 / S 1 * * . 5) 137 S E 1 = S E - F F / F F 1 1 4 0 E R R » A B S ( S E 1 - S E ) 1 4 1 I F ( E R R . G T . 0 . 0 0 0 1 ) G 0 T O 32 1 4 2 S E ^ S E I 1 4 3 S A V « 2 4 0 . / ( 1 9 1 . / S H + 2 6 » 5 / S V + 2 2 . 5 / S E 1 4 4 V L » ( 1 3 . - 2 7 . * D P R ) 1 4 5 E L = ? . 8 5 + 7 . 1 2 * L P R 1 4 6 H L » 1 9 1 . - 2 0 7 . * D P R 1 4 7 C F 1 » . 1 6 4 * F 1 * E H 0 * V A 3 C * * 2 150 C F 2 = ( 1 0 7 . * A S P ) / W M 151 C F 3 = 7 . 0 * F 4 * W M * S A V * V A 3 C 152 D I M E N S I O N A ( 1 0 , 4 ) , B ( 5 , 4 ) , X ( 1 2 0 , 3 ) , X S U M ( 3 ) , X M E A N ( 3 ) , X S T D ( 3 ) , S T D D ( 3 ) 153 A ( I , 1 ) « ( D P / 2 . 0 3 6 - C F 1 * ( 1 o + D P R ) ) * C P 2 154 A ( 1 , 2 ) » H L * G / ( S H * V A 3 C ) + E L * S E * V A 3 C 155 A ( 1 , 3 ) = 7 6 0 . * S A V * V A 3 C * ( 1 . + D P R ) 1 5 6 A ( 1 , 4 ) - S E * V A 3 C * ( 1 . + 2 . * D P R ) + G * V L / ( S V * V A 3 C ) 1 5 7 R 1 = X N / X N O 1 6 0 R 2 = ( X N 0 - X N ) / X N 1 6 1 R 3 « ( X N 0 - X N ) / X N 0 1 6 2 R 4 = P 3 0 / P 2 1 0 1 6 3 R5=P3/P21 1 6 4 R 6 = ( 1 . + X N * * 2 * R 5 * * ( 2 . / X N ) + 2 . * X N * R 5 * * ( 1 . / X N ) ) / ( X N + 1 . ) * * 2 1 6 5 R 7 - R 5 * ( 2 . / X N ) 1 6 6 R 8 = R 6 / R 7 188 167 R9=(DP-DPO)/P30 170 R10=(DP~DP0)/P3 171 R11=(CF1*2.036)/DP 172 R12=(DP-2.036*CF1)/DP 173 R13=CP2/DP 17A- R14=(WA/WA0 ) * ( VA3C0/VA3C ) 175 R15=((WA+WM)/WA0)*(VA3C0/VA3C) 176 R16=R14*AMR 177 R17=R15*AMR 200 R18=(WA/WA0)* AMR 201 R19=(VAAV*SAV) 202 R20=WM/WA 203 R21=CD/REP 204 R22=CD*REP 205 R23=(CD*SEP**2)**o3333 206 R24=(REP/CD)* *.3355 207 WRITE(6 ?8) 210 8 FORMAT (5X,5HDPO, 6X,5HVTSC0, 6X,4HRH00, 7X, 3HWA0, 6X, 4EVA30 1 6X,5HVA3CO, 7X,3HREO,6X,3HXNO, 6X,3HF10,//) 211 WRITE (6,9; DPO, VISC0,RH00, WAO, VA50, VA5C0, REO, XNO, F10 212 9 FORMAT (2X,F8.4,E12.4,4F10.4,E12.4,2F10.4,///) 215 WRITE (6,10) 214 10 FORMAT (5X,2HDP,7X,4HVISC,7X,5HRH0,8X,2HWA,7X,5HVA3, 1 7X,4HVA30,8X,2HRE,7X,2HXIT,7X,2HF1,5X,4HYAAV,5X,4HVMAV,//) 215 WRITE(6,11)DP,VISC,RH0,WA,VA5,VA3C,RE,XN,F1,VAAV,VMAV 216 11 F0RMAT(2X,F8,3,E12.4,4F10.4,E12.4,4F10.4,///) 217 WRITE (6,13) 220 13 F0RMAT(6X,3HAMR,8X,5HGAMMA,10X,2HWM,10X,1HS,10X,1HA,11X, 1 11X,1HC,11X,1HD,9X,4HTAMX,9X.4HVMMX,//) 1HB, 221 WRITE (6,14)AMR,GAMM,WM,S,A(I,1),A(Ii2),A(I,5),A(I,4), VAMX,VMMX 222 14 FORMAT (2X,4F12.6,4F12.2,2F12.4///) 225 WRITE (6,16) 224 16 FORMAT (3X,3HCF1,7X,3HCF2,7X,2HR2,8X,2HR3,8X,2HR4,8X,2H5 1 8X,2HR6,8X v 2HR7,8X,2HR8,8X,2HR9,//) 225 WRITE(6,17)CF1,CF2,R1,R2,R3,R4,R5,R6,R7,R8,R9 226 17 FORMAT (F10.4,F10.1,9F10.4,///) 227 WRITE (6,18) 230 18 FORMAT (3X,3HR10,7X,3HR11,7X,3HR12,7X,3HR13,7X,3HA/B, 7X,3HA/C,7X, 1 3EA/D,7X,3HB/C,7X,3HB/D,7X,3HC/D,7X,3HR20,//) 231 WRITE (6S17)R10,R11,R12,R13,R14,R15,R16,R17,R18,R19,R20 232 WRITE (6,70) 235 70 FORMAT (7X,2HDD,9X,3HDM2,10X,2HCD,13X,5HREP,12X,5HR21, 12X,5HR22, 1 12X,3HR25,12X,5HR24///) 234 WRITE (6,71) DD, DM2,CD, REP,R21,R22,R23,R24 235 71 FORMAT (3F12.5,5E15.5///) 236 X3=F3 237 X4=F4 240 X2=(A(1,2)*X3+A(1,5)*X4+A(1^,4))/(A(1,1)-A(1,3)*X4) 241 242 22 243 • 244 23 245 15 C 246 24? 50 250 251 252 51 253 52 254 108 255 256 257 100 260 261 262 103 263 264 265 266 267 270 105 271 272 273 274 275 276 277 300 106 301 107 302 303 57 304 305 306 307 310 311 C 312 313 314 315 316 120 317 189 WRITE (6,22) FORMAT (5X,5HRH0AV,5X,2HX2,8X,2HX3,8X,2HX4,8X,2HSH,8X, 2HSV,8X,2HSE, 7X,3HCF3,7X.3HSAV/) WRITE (6,23) RH0AV,X2,X3,X4„SH,SV,SE,CF3,SAV FORMAT (3F10.5,E10„5,6F10.5) CONTINUE WRITE (6,50) FORMAT (1HI,16X,1HE,4X,1HL,4X,1HM,10X,5HCD/F2,10X, 2HF3,12X,2HF4,//) IJ=0 K=1 L=K+1 M=L+1 DO 100 J=1,4 B(1,J)=A ( K,J) B(2,J)=A(L 9J) B(3,J)=A(M,J) DO 103 JJ=1,3 DO 103 J=2„3 B ( J J , J ) = - B ( J J , J ) DO 105 KK=1,2 KP=KK+1 DO 105 11=KP,3 RF=B(11 , K K ) / ( B ( K K : , K _ ) DO 105 J=KK,4 B(11 , J ) = B(11,J)-RF * B(KK,J) IJ=IJ+1 X(IJ,3)=B(3,4)/B(3,3) DO 107 KK=1,2 II=3-KK IP=11+1 S - B(11,4) DO 106 J=IP,3 S = S - B(11,J)*X(IJ,J) X(IJ,11 ) = S / B(11,11) WRITE (6 , 57 ) K,L , M,X(IJ , 1),X(IJ,2),X(IJ , 3 ) FORMAT (13X,315,3E15.5//) M=M+1 IF(M.LE.IO) GO TO 108 L=L+1 IF(L.LE.9)G0 TO 52 K=K+1 IF(K.LE.8)G0 TO 51 DO 122 1-1,3 XSUM(1)=0.0 XSTD(1)=0.0 DO 120 J=1,120 XSUM(1)ixSUM(1)+ X(J,I) XMEAN(I)=XSUM(I)/120 190 320 DO 121 K=1,120 321 121 XSTD(I)=XSTD(I)+(X(K,I)-XMEAN(I))**2 322 122 STDD(I)=SQRT(XSTD(I)/120.) 323 WRITE (6,58) 324 58 FORMAT (1H1,13X,8HCD/F2BAR,12X,5HF3BAR,15X,5HF4BAR,15X 9HCD/F2STDD, 1 11X,6HF3STDD,14X,6HF4STDD//) 325 WRITE (6,59) XMEAN(1), XMEAN(2),XMEAN(3),STDD(1), STDD(2),STDD(3) 326 59 FORMAT (6E20.5) 327 MM=MM+1 330 IF(MM„LE.3) GO TO 20 331 F4=F4+0o025E-02 332 IF(F4.LE.0.325E-02)G0 TO 200 333 STOP 534 END SENTRY 

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